Robustness: a New Form of Heredity Motivated by Dynamic Networks

05/10/2019 ∙ by Arnaud Casteigts, et al. ∙ Laboratoire d'Informatique de Paris 6 0

We investigate a special case of hereditary property in graphs, referred to as robustness. A property (or structure) is called robust in a graph G if it is inherited by all the connected spanning subgraphs of G. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion ( i.e. they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness. We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of maximal independent sets (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs in which all MISs are robust. Then, we turn our attention to the graphs that admit a robust MIS (). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists.

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1 Introduction

The area of dynamic networks covers a variety of contexts, ranging from nearly-static networks where the network topology changes only occasionally, to highly-dynamic settings where the entities interact in a volatile way, through communication links which appear and disappear arbitrarily often and unexpectedly. In the second case, the immediate structure of the communication graph at given time (i.e., its structural snapshot) does not capture much information, the main features being rather of a temporal nature. For example, the snapshots may never be connected, and yet offer a form of connectivity over time and space, called temporal connectivity. A number of formalisms were proposed recently to capture the temporality of these contexts, such as evolving graphs, time-varying graphs, link streams, and temporal graphs (see e.g. Fer04 ; CFQS12 ; LVM17 ; KKK02 , among many others). In this article, we introduce a graph concept which is strongly motivated by the highly-dynamic setting; however, the notion itself can be formulated and studied in terms of standard graphs, independently from its temporal interpretation.

Given a (standard) graph , a given property (or structure) in is said to be robust if and only if it is inherited by every connected spanning subgraph of . Hereditary properties based on the removal of edges are generally called monotone (see for instance heredity1 ; heredity2 ; K88 ). Robustness is therefore a particular case of monotonicity, in which the subgraph is additionally constrained to remain spanning and connected. (This concept is different from other uses of the term “robustness” in the literature, see e.g. robustness3 ; robustness1 ; robustness2 .)

As explained, robustness can be interpreted in several different ways. (1) Static networks with permanent link crashes: Here, the network is essentially static; however, it deteriorates over time and some of the links may definitively stop working (or equivalently, be removed). The network is to be used so long as it still connects all the nodes. It is easy to see, that if a property is robust in such a network, then it will remain valid so long as the network is used, despite the uncertainty regarding which of the links will crash. (2) Recurrent temporal connectivity in highly-dynamic networks: Here, the immediate structure of the network does not matter as much as its temporal properties. In particular, a basic assumption is that temporal paths exists recurrently between all the entities, corresponding to Class  in Cas18 and Class 5 in CFQS12 . Dubois et al. observed in DKP15 that this assumption is equivalent to the guarantee that a subset of the edges always reappears, although in general such as subset is not known in advance. Thus, robustness can be interpreted in as the fact that a property is satified with respect to the subset of recurrent edges, whichever these edges are. The choice for a given interpretation is not mandatory, as robustness can be studied independently. However, the second interpretation being the original motivation here, we develop it further in a dedicated (but optional) section in the end of the paper.

In this paper, we illustrate the concept of robustness through a classical covering problem called maximal independent set (MIS), which consists of selecting a subset of vertices none of which are neighbors (independence) and which is maximal for inclusion. Let us first observe that robust MISs may or may not exist depending on the considered graph. For example, if the graph is a triangle, then only one such structure exists up to isomorphism, consisting of a single vertex (refer to Figure 1). If an edge next to the selected vertex is removed, then this set is no longer maximal. Therefore, the triangle graph admits no robust MIS.

(a)

(b)

(c)

(d)
Figure 1: Four examples of MISs in various graphs.

Some graphs admit both robust and non-robust MISs, as exemplified by the bull graphs on Figures 1 (non-robust) and 1 (robust). Finally, some graphs like the square graph (Figure 1) are such that all MISs are robust.

1.1 Contributions

In addition to the concept itself and the related discussions, we characterize exactly the set of the graphs in which all MISs are robust. To this end, we first define a class of graphs called sputniks (for reasons that will become clear later). Sputniks include, among others, all the trees, for which every property is trivially robust (since none of the edges are removable). We show that consists exactly of the union of sputniks and complete bipartite graphs. The interest of this universal class is that finding a robust MIS in it amounts to finding a standard MIS.

The existential versions of this class, namely the set of those graphs which admit a robust solution seem to have a more complex structure. We first give a sufficient condition and we show that this condition is necessary in the particular case of biconnected graphs (meaning here 2-vertex-connected). The more general case is addressed by means of an algorithm that decides if a given graph belongs to . The trivial strategy for such an algorithm would amount to enumerating all MISs until a robust one is found. However, exponentially many MISs may exist in general graphs (see Moon and Moser MM65 , and F87 ; GGG88 in the particular case of connected graphs) and the validity of each one may have to be satisfied in exponentially many connected spanning subgraphs. Motivated by this observation, we present a polynomial time decision algorithm, which can be adapted into a constructive algorithm (without significant overhead). Our algorithm relies on a particular decomposition of the graph into a tree of 2-vertex-connected (biconnected) components called an -tree (a variant of block-cut trees Harary ), along which constraints are propagated as to the MIS status of intermediate vertices. The inner constraints of non-trivial components are solved by a reduction to the 2-SAT problem. The yes-instances of this algorithm characterize , albeit indirectly. Whether admits a more elementary characterization in terms of graph properties is left open.

1.2 Organization of the paper

Section 2 presents the main definitions and concepts. Next, Section 3 presents the characterization of class . In Section 4.1, we show that if the graph is biconnected, then being bipartite is both necessary and sufficient to belong to . Then, we present the decision algorithm in Sections 4.2 through 4.6, its complexity in Section 4.7 and its adaptation into a constructive algorithm in Section 4.8. In Section 5, we develop the discussion regarding the temporal interpretation of robustness, motivated by highly-dynamic networks. We conclude in Section 6 by observing some additional features of robustness in general and a few open questions.

2 Main Concepts and Basic Results

Let be a simple connected undirected graph on a finite set of vertices (or nodes). We denote by the neighbors of vertex , i.e., the set . The degree of a vertex is . A vertex is pendant if it has degree . An articulation point (or cut vertex) is a vertex whose removal disconnects the graph. A bridge (or cut edge) is an edge whose removal disconnects the graph. We say that an edge is removable in a graph if it is not a bridge of . Given , the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in . In the context of this paper, a graph is said to be biconnected if it has at least three vertices, and it remains connected after the removal of any single vertex (i.e., 2-vertex-connectivity). A biconnected component is a maximal biconnected subgraph. Finally, a spanning connected subgraph of a graph is a graph such that , and is connected. We define the concept of robustness as follows.

Definition 1 (Robustness)

A property is robust in if and only if it is satisfied in every connected spanning subgraph of (including itself).

In other words, a robust property holds even after an arbitrary number of edges are removed without disconnecting the graph. Robustness is a special case of hereditary property, and more precisely a special case of a decreasing monotone property (see for instance K88 ). The term “property” includes both basic graph properties and solutions to combinatorial problems. Our focus in this initial work is on the latter; however, looking at the robustness of basic graph properties might help understand this notion further, for instance, connectivity itself is a trivial robust property. Bipartiteness is also a robust that is discussed at several occasions in this paper.

In the present work, we focus on the maximal independent sets (MIS) problem, which consists of selecting a subset of vertices none of which are neighbors (independence) and to which no further vertex can be added (maximality). Following Definition 1, a robust MIS (RMIS, for short) in a graph is a subset of vertices which remains maximal and independent in every connected spanning subgraph of .

Observation 1

The notion of independence is stable under the removal of edges. Therefore, it is sufficient that an MIS remains maximal in order to be an RMIS.

Let us define the following two classes of graphs.

Definition 2 ()

Set of graphs in which all MISs are robust.

Definition 3 ()

Set of graphs that admit at least one robust MIS.

Trivially, . Finally, we define two classes of graphs which play a central role in this work, namely complete bipartite graphs and sputnik graphs, the latter being new.

Definition 4 (Complete bipartite graph)

A complete bipartite graph is a graph such that and .

In other words, the vertices can be partitioned into two sets and such that every vertex in shares an edge with every vertex in (completeness), and these are the only edges (bipartiteness).

Definition 5 (Sputnik)

A graph is a sputnik if and only if every vertex belonging to a cycle has at least one pendant neighbor.

An example of sputnik is shown on Figure 2. This name was chosen by analogy with the well-known satellite. By the same analogy, we say that a vertex has an antenna if it has at least one pendant neighbor.

Figure 2: Example of a sputnik graph (whose planarity is accidental).

3 Characterization of

In this section, we show that the class of graphs in which all MISs are robust, , corresponds exactly to the union of complete bipartite graphs and sputnik graphs. We first show that all the MISs in a complete bipartite graph are robust, and so are all MISs in a sputnik graph (sufficient conditions). Next, we show that these graphs are the only ones (necessary condition). The necessary side is more complex because two classes of graphs are involved. The proof proceeds by showing that if all MISs are robust in a graph which is not a sputnik, then that graph must be a complete bipartite graph.

Lemma 1

All MISs in a complete bipartite graph are robust.

Proof: From Observation 1, we only need to show that maximality is preserved. There are two ways of chosing an MIS in a complete bipartite graph , namely or . Without loss of generality, let be chosen. Then, in all connected spanning subgraphs of , every vertex in still has at least one neighbor in (otherwise the graph is disconnected), which preserves maximality.

Lemma 2

All MISs in a sputnik graph are robust.

Proof: Again, by Observation 1, we only need to show that maximality is preserved. By definition of a sputnik graph (Definition 5), if an edge is removable (i.e., it belongs to a cycle), then both of its endpoints have an antenna. Maximality implies that either or the tip of its antenna are in the set. (The same holds for .) As antennas are bridges, they cannot be removed and thus maximality is preserved when edges from a cycle are removed.

Lemma 3

If is not a sputnik, and every MIS in is robust, then must be a complete bipartite graph.

Proof: If is not a sputnik, then some vertex belonging to a cycle has no antenna. Consider the graph and call the components that would result, except that a copy of is added back to each (see Figure 3 for an illustration). Without loss of generality, let be the one containing . The other components (if any) are such that every neighbor of has another neighbor than (otherwise would have a pendant neighbor).

Claim. If all MISs in are robust, then all neighbors of in have the same set of neighbors.

We prove this claim by contradiction. Let two neighbors of be such that . We will show that at least one MIS is not robust (see Figure 3 for an illustration). Without loss of generality, let be a vertex in . Then an MIS can be built which contains both and (as a special case, may be the same vertex as , which is not a problem). For each of the components , choose an edge and add another neighbor of to the MIS (such a neighbor exists, as we have already seen). One can see that , and all can no longer enter the MIS because they all have neighbors in it. Now, choose the remaining elements of the MIS arbitrarily. Now, consider the following removals: in all components all edges incident to except are removed; and in , all edges incident to except are removed. The resulting graph is connected since all of the are connected, but the set is no longer maximal because could now be added in it. (End of Claim)

Figure 3: Contradictory example for proving the claim, where . The dashed line represents the rest of component .

Since ’s neighbors in have the same neighbors, this means in particular that none of these vertices has an antenna. As a result, the arguments that applied to because of its absence of antenna, apply in turn to ’s neighbors in . In particular, the neighbors of these vertices (including ) must have the same set of neighbors. This implies that (1) can no longer be an articulation point, thus and (2) all neighbors of have the same set of neighbors, and these neighbors also have the same set of neighbors, which finally implies that the graph is complete bipartite.

Lemmas 1, 2, and 3 allow us to conclude with Theorem 1 :

Theorem 1

All MISs are robust in a graph if and only if is complete bipartite or sputnik.

4 Characterization of

In this section, we turn our attention to the characterization of graphs which admit a robust MIS (). Unfortunately, this class does not seem to admit a simple characterization in terms of elementary graph properties. We start by discussing some such properties that give a partial characterization, namely we show that bipartiteness is a necessary and sufficient condition for biconnected graphs to be in . Then, we turn our attention to the general case and present an algorithm that decides if a given graph admits a robust MIS (in polynomial time, and constructively). The yes-instances of this algorithm are an indirect characterization of .

4.1 Bipartiteness versus Biconnectivity

Let us first observe that bipartiteness is a sufficient condition for any graph to admit a robust MIS.

Lemma 4

All bipartite graphs admit a robust MIS.

Proof: The argument generalizes that of Lemma 1 for complete bipartite graphs. Let an MIS be composed of all the vertices of the first part. As long as the graph remains connected, every vertex in the second part has a neighbor in the first part, and thus in the MIS, which preserves maximality. (Independence is not impacted—Observation 1.)

In fact, bipartiteness happens to be also a necessary condition in the particular case of biconnected graphs.

Lemma 5

If a biconnected graph is not bipartite, then it admits no robust MIS.

Proof: By contradiction, suppose that admits a robust MIS. As is not bipartite, it is not 2-colorable, thus either two neighbor vertices exist which are both included in the set, or two neighbor vertices exist which are both excluded from the set. In the first case, independence is contradicted. In the second case, let and be two such vertices. Because the graph is biconnected, it is possible to remove all the incident edges to except without disconnecting , resulting in a non-maximal set, thus contradicting robustness.

The argument in the proof of Lemma 5 will be used several times in the rest of the paper. We refer to it through the concept of a weak vertex.

Definition 6 (Weak vertex)

Let be a vertex that is not included in an MIS. If has another neighbor not being included in the MIS and such that all edges incident to except can be simultaneously removed without disconnecting the graph, then is called a weak vertex.

Lemmas 4 and 5 imply that the intersection of with biconnected graphs is exactly the set of bipartite biconnected graphs (see Table 1).

Bipartite Bipartite
Biconnected yes no
Biconnected yes possibly
Table 1: Existence of a robust MIS with respect to bipartiteness and biconnectedness.

An immediate corollary is the existence of an infinite family of graphs which do not admit a robust MIS, namely all biconnected non-bipartite graphs (including, for example, the triangle graph in Figure 1

and more generally all the odd cycles).

4.2 Overview of the algorithm

The problem of computing a standard MIS in a graph can be solved by a one-sentence greedy algorithm as follows: for all vertices in arbitrary order, include in the MIS if none of its neighbors already is. The problem of computing a robust MIS (or RMIS) is fundamentally different in two respects: (1) Solutions may or may not exist, and (2) Even if a solution exists, a decision made in some part of the graph can restrict (or invalidate) the feasible choices in remote parts. For example, in the graph of Figure 4, if vertex is included in the set, then an RMIS exists if and only if node is not included in the set. (Any other choice would produce a weak vertex.)

Figure 4: Non-locality of finding a robust MIS.

More generally, deciding whether a graph admits an RMIS requires the identification and propagation of constraints within the graph. To do so, our algorithm relies on a particular type of decomposition of the input graph as a tree of biconnected components called -tree. The constraints are first determined at the leaves of this tree, then they are propagated and modified upward, until the root component is itself analysed. At an intermediate node of this tree, either the corresponding subtree admits an RMIS or it does not. If it does, a condition may apply regarding the status of the topmost vertex in the subtree, e.g. the subtree may admit an RMIS at the condition that this vertex is (or is not) in the MIS.

In the following, we always refer to the vertices of the input graph as vertices, and to those of the decomposition tree as nodes to avoid confusion. Note that if is a tree, then none of its edges can be removed (thus all MISs are robust). As a result, we restrict our attention to the cases that has at least one biconnected component. We also assume that is connected, as otherwise each part can be solved separately.

4.3 The -tree decomposition

An -tree, denoted by , is neither a block-cut tree, nor a bridge tree (see Harary for background), it is a mixture of both. Precisely, an -tree is made of four types of nodes and defined with respect to the input graph as follows:

  • is the set of pendant vertices,

  • is the set of articulation points,

  • is the set of bridges,

  • is the set of biconnected component.

Thus, every node in and corresponds to an original vertex in (the converse is not true), every node in corresponds to an edge in (same remark), and every node in corresponds to an entire subgraph of  (same remark again). Considering the graph in Figure 5, one would obtain , , , and .

Observe that a same vertex of may correspond to a node , and at the same time be the endpoint of one or several bridges in , and at the same time belong to one or several biconnected components in . Also observe that the endpoints of a bridge are always articulation points or pendant vertices. All these relations are materialized by the set of edges of the -tree, defined as

.

In words, articulation points (seen as nodes) share an edge with the biconnected components they belong to (if any), and articulation points or pendant nodes (seen as nodes) share an edge with the bridges they belong to (if any). The -tree corresponding to the graph of Figure 5 is shown in Figure 6. The reader is encouraged to spend a few minutes getting acquainted with this construction, which is used frequently in the following.

Figure 5: Example of input graph, seen as a tree of biconnected components.

: pendant vertices () : articulation points () : bridge edges () : biconnected components ()
Figure 6: -tree corresponding to the input graph on Figure 5.

4.4 RMIS constraints in a rooted -tree

The algorithm proceeds by propagating constraints from the leaves of the -tree to a root chosen arbitrarily among the biconnected components. Given a node in , we denote by the neighbor of that is one-hop closer to , and by the set of nodes . We extend these definitions to in the natural way. Because some nodes in correspond to real vertices (namely, the nodes in and ) and others do not (the nodes in and ), we define a notion of attachment vertex of a node as being either the underlying vertex itself (if ) or the underlying vertex of the parent (if ). Indeed, the parent of a node in is always a node in , thus it corresponds to a real vertex in . Intuitively, the attachment vertex of a node is the highest vertex in this node towards the root . For instance, in Figure 6, assuming that is component D, then we have for instance , , , , and , and indeed, is the highest underlying vertex in all these nodes (with respect to the root).

Attachment vertices play a important role in the management of constraints, because the constraints induced by are ultimately aggregated as a membership status for in the MIS. The more formal definition relies on induced subgraphs as follows. Every node induces a subgraph which is exactly when (see definitions in Section 2), or itself when . By extension, for any , the subtree whose highest node with respect to is induces a subgraph defined as over all . In other words, is the subgraph of which corresponds to the subtree of . We define the concept of aerial version of , noted as with an artificial neighbor of that does not belong to . For any , the constraints applying to are encoded using the three following labels (also called tags):

  • PI (Possibly In): admits an RMIS that includes ;

  • PO (Possibly Out): admits an RMIS that does not include ;

  • PE (Possibly External): admits an RMIS that does not include .

Observe that a node may have several labels at the same time. For example PI and PO, or PI and PE. On the other hand, we use PO and PE in a mutually exclusive way based on the following remark.

Remark 1

If a node has label PO, then admits an RMIS in which is not included but one of its neighbor in is (due to maximality). Thus, no inclusion constraint applies regarding the external neighbor of . As a result, whenever the constraints from different children induce both PO and PE, PO is chosen.

4.5 Constraint identification and propagation

In this subsection, we present the rules used for identifying and propagating constraints within the -tree. The purpose of the rules is to determine what labels a node should take based on the labels of its children in the -tree. The validity of the rules is established gradually along their descriptions, based on the following definition of a correct labeling.

Definition 7 (Correct labeling)

A node (in the underlying rooted -tree ) is correctly labeled if

  • if and only if admits an RMIS that includes ;

  • if and only if admits an RMIS that does not include ; and

  • if and only if and admits an RMIS that does not include .

The rules are presented below based on the type of (namely, and ). They are illustrated in reference to the input graph in Figure 5 and the corresponding -tree in Figure 6, with root . For simplicity, when discussing about the construction of an MIS, if a vertex is not included in the MIS and none of its neighbor are included, we say that is not covered. The other vertices are covered. (This terminology is standard in the literature on covering problems.)

4.5.1 Pendant nodes ()

This case is the easiest, because nodes in have no children and their labels are always the same.

Labeling rule 1

is set to .

In words, may or may not be included to the MIS, but if it is not, then should be, where is the bridge node such that .

Lemma 6

If , then Labeling rule 1 produces a correct labeling of .

Proof: If is included in the solution set, then this set (made of alone) is clearly maximal and independent in (which is also alone), thus can be labeled PI. If it is not included, but is included, then the resulting set is maximal and independent in (i.e., the graph made of a single edge between and ), thus can be labeled PE. In both cases, the considered subgraph of is itself a tree, thus any valid MIS in it is robust, thus labels PI and PE are both valid. On the other hand, cannot be labeled PO because if is not in the MIS, then the corresponding set is not maximal in (which is reduced to the single vertex ).

4.5.2 Articulation points ()

By construction of the -tree, if , then all the children of are in or in and their attachment vertex is nothing but itself. Thus, the constraints of each children already relates to , albeit individually. The rule consists of aggregating these constraints as follows.

Labeling rule 2

If PI belongs to the labels of all the children of , then PI is added to ; if all the children of contain a label PE or PO, then two subcases arise: either none of them contains PO, in which case PE is added to , or at least one contains PO, in which case PO is added to .

The formal description of Labeling rule 2 is given by Lines 6 to 13 in Algorithm 1 (the algorithm itself is discussed in a subsequent section). In our example, nodes , , and are all labeled PI and PO.

Lemma 7

If and all the nodes in are correctly labeled, then Labeling rule 2 produces a correct labeling of .

Proof: Let us assume that all the nodes in are correctly labeled. The proof follows the same cases as the labeling rule. We first prove that the assigned labels are valid, then we prove that they cannot be assigned otherwise.

  • If all the nodes in contain label PI, then for all , admits a robust MIS that includes . But since , we have , thus admits a robust MIS that includes .

  • If all the nodes in contain label PE or PO, then two possible subcases arise:

    1. None of them contains PO. In this case, all of them contain label PE, meaning that for all does not admit a robust MIS that excludes , but does. Since , we have that does not admit a robust MIS that excludes (a single child with label PE would actually be enough here), but does (here, we need that all the children have label PE).

    2. At least one contains PO. In this case, at least one child is such that admits an RMIS that excludes . Since , at least one of the neighbors of in can be included in such an MIS, which satisfies the aerial constraints of the for all other children labeled PE (if any). Thus admits an RMIS that excludes without requiring further aerial constraints above .

We focus now on the negative direction. If does not admit a robust MIS that includes , then because , there exists at least one such that does not admit a robust MIS that includes . As the labeling of is correct, is not labeled PI and hence is not labeled PI by the rule. Similarly, if does not admit a robust MIS that excludes , then because , the two cases above are possible. There exists at least one such that neither nor admit a robust MIS that excludes . As the labeling of is correct, is not labeled PO or PE. For any , does not admit a robust MIS that excludes . As the labeling of is correct, is not labeled PO. In both cases, cannot be labeled PO by the rule. Finally, if does not admit a robust MIS that excludes , then because , there exists at least one such that does not admit a robust MIS that excludes . As the labeling of is correct, is not labeled PE and hence is not labeled PE by the rule.

4.5.3 Bridge nodes ()

If is a bridge node, then it has exactly one child whose attachment vertex is a neighbor of . The rule consists of transforming the existing constraints on into constraints on as follows.

Labeling rule 3

If contains label PI, then label PO is added to ; if contains label PE, then PI is added to ; if contains label PO, then PI and PE are added to . Finally (cleaning), if both PE and PO have been added to , then PE is removed from (see Remark 1).

The formal description of Labeling rule 3 is given by Lines 15 to 23 in Algorithm 1. In our example, the nodes , , and are all tagged PI and PO.

Lemma 8

If and is correctly labeled, then Labeling rule 3 produces a correct labeling of .

Proof: Let and .

  • If contains label PI, then admits a robust MIS including . Because is included, the MIS is also valid in , and because this edge is a bridge (i.e., it is not removable), it remains robust in , thus can be labeled PO.

  • If contains label PE, then admits a robust MIS including , thus can be labeled PI.

  • If contains label PO, then admits a robust MIS that excludes .
    - As is the only neighbor of in , adding to such an MIS would produce a valid MIS in , and because is a bridge, the resulting MIS would remain robust. Thus can be labeled PI.
    - As already has a neighbor included in the MIS in , not including in the MIS means that is the only uncovered vertex in , which can be remedied by including to the MIS an aerial neighbor in . Thus, can be labeled PE.

We focus now on the negative direction. If does not admit a robust MIS that includes , then because is not a removable edge, does not admit a robust MIS that excludes and does not admit a robust MIS that excludes (and includes ). As the labeling of is correct, is not labeled PI or PE and hence is not labeled PI by the rule. Similarly, if does not admit a robust MIS that excludes , then because is not a removable edge, does not admit a robust MIS that includes . As the labeling of is correct, is not labeled PI and hence cannot be labeled PO. Finally, if does not admit a robust MIS that excludes , then because is not a removable edge, does not admit a robust MIS that excludes . As the labeling of is correct, is not labeled PO and hence is not labeled PE by the rule.

Parameters: A node whose children in are already labeled
Return: A correct labeling of 01: 02: 03: case : 04: 05: 06: case : 07: if then 08: 09: if or then 10: if then 11: 12: else 13: 14: 15: case : (with child side ) 16: if then 17: 18: if then 19: 20: if then 21: 22: if and then 23: 24: 25: case : 26: if isSatisfiable then 27: 28: if isSatisfiable then 29: 30: else 31: 32: if isSatisfiable then 33: 34: 35: 36: return

Algorithm 1 LabelNode

4.5.4 Biconnected components ()

As discussed in Section 4.1, when itself is a biconnected component, an RMIS exists if and only if is bipartite (Lemmas 4 and 5). When a biconnected component is a node in the -tree , the situation is more complex due to the existence of constraints from neighbors in . In particular, an RMIS satisfying these constraints may or may not exist.

Let be the subgraph of induced by node (in fact, coinciding with ). By construction of the -tree, every neighbor of in (whether children or parent) corresponds to an articulation point in such that . Thus, potential constraints on these nodes can be seen as applying to vertices inside . This nice feature allows us to focus on finding an RMIS in without worrying about the entire subtree . Indeed, if an RMIS satisfying the constraints exists in and if every child is correctly labelled, then an RMIS must exist in every such that has the same status (included or excluded) in and in . As every is an articulation point, the union of all these RMISs forms a robust MIS in .

Based on the above observation, we now focus on the simpler problem of finding an RMIS in that satisfies potential constraints on its articulation points. In addition, we add an input parameter to the problem that forces the status of the attachment vertex in the RMIS, so that (for instance) receives label PI if an RMIS is found that includes . (The exact labeling rule using will be described later on.)

4.5.4.1  Solving

Here, we consider the following problem. Given a biconnected component , some vertices of which are articulation points in (corresponding to nodes in ); given constraints on these articulations points, encoded into the labeling function ; given a parameter or ; the goal is to determine if there exists a robust MIS in in which is included in the RMIS if the parameter is , not included if the parameter is , and unrestricted if the parameter is , and such that the inclusion status of every constrained articulation point matches at least one of its labels. Namely, if (i.e., is constrained), then can be included to the RMIS only if contains PI; and it can be excluded from the RMIS only if contains PO or PE.

Bipartiteness vs. Po labels

As already explained, if a biconnected graph is not bipartite, then weak vertices must exist (see Lemma 5 and Definition 6). Bipartiteness still plays a central role in the case of biconnected components, but the existence of PO labels makes it more subtle, as explained in the following remark.

Remark 2

Let be an articulation point such that and contains label PO, then there exists an RMIS in that does not include itself but includes one (or possibly several) neighbors in that prevent from being a weak vertex.

As a result, the vertices admitting a label PO are not subject to the same bipartiteness constraint as the others vertices.

The procedure

The procedure is formally described in Algorithm 2, to which the reader is referred for details, and its correctness is proved subsequently. Let be the set of edges in such that both endpoints contain label PO, and let be the subgraph , possibly resulting into several connected components . If is not bipartite, then the algorithm rejects, because this means that at least one non-PO vertex must be weak. Otherwise, the component may possibly admit an RMIS if the combination of constraints induced by all the vertices is satisfiable. The rest of the procedure consists of encoding these constraints into a 2-SAT formula such that admits an RMIS (with given status for , the attachment vertex), if and only if the formula is satisfiable. The formula is built as follows. For each connected component , a SAT variable is created. As is bipartite, every vertex of one part receives label and every vertex of the other part receives a label . (Intuitively, if the eventual formula is satisfiable with , then the vertices in the first part are included in the RMIS; if it is satisfiable with , then the vertices of the second part are included.) Then, the existing constraints of articulation points are incorporated; namely, if an articulation point contains only label PI, then is set to true; if it contains only PO or only PE, then is set to false; if it contains both options (i.e., PI PO or PI PE), then no constraint is added. Finally, although the edges of induce no constraints for bipartiteness, we must still make sure that their endpoints are not both included to the RMIS (for independence), thus if , then the clause is added.

Parameters: A node and a that forces the status of
Return: if a suitable set of exists, otherwise 01: Let be the set of edges of such that and 02: Let 03: if is not bipartite then 04: return false 05: Let be the (bipartite) connected components in 06: Let be an empty -SAT expression on a set of boolean variables 07: foreach in do 08: Assign to each vertex in a label such that neighbors have different labels 09: foreach do 10: if then 11: 12: if or then 13: 14: foreach do 15: 16: if then 17: 18: else if then 19: 20: return whether is satisfiable (using an external 2-SAT solver)

Algorithm 2 isSatisfiable
Correctness

To formalize the properties of the procedure , we need to introduce some definitions. Let be the graph built from by adding a small gadget to every when is constrained (i.e., is non-empty). The gadget consists of a path of length incident to through virtual vertices and (see Figure 7 for an intuition). This construction is not used in the procedure itself, only in the proof.

1

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PI PO

PI PO

PI PO

PI

B

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Figure 7: Example of component (left)– corresponds to the component B in Figure 5. With , a chain of two virtual nodes is added to each constrained node that gives the corresponding (right). The MIS shown on corresponds to the set used in the proof of Lemma 9 with included.

We say that a set of vertices is a suitable RMIS of if and only if is an RMIS of such that, for every with at least one label in , we have:
- in only if PI ;
- not in only if PO or PE appears in ; and
- not in , in , and not in only if PO appears in .

The idea is that, for every , the graph induced by the path replaces in the management of constraints so that a suitable RMIS of exists if and only if an RMIS of exists. We now prove the main lemma.

Lemma 9

Given a biconnected component with constraints on some articulation points, returns true if and only if a suitable RMIS of including exist; returns true if and only if a suitable RMIS of excluding exist; returns true if and only if a suitable RMIS of exists, irrespective of .

Proof: Let be the considered node in the -tree and the corresponding component. Let be the augmented version of as described above. First observe that, if is bipartite, then so is , because the adjunction of such paths cannot affect bipartiteness. In the following, we first prove the direction that (1) if the procedure returns true, then the corresponding type of suitable RMIS exists; then (2) if the given type of suitable RMIS exists, then the procedure with corresponding parameters must return true.

  1. If is satisfiable, define as the set of vertices induced by a positive assignment, i.e., . Now, add to some of the virtual vertices as follows: if is in , add only ; if is not in and is weak in , add only ; if is not in and is not weak in , add only . Lines 16 to 19 ensure that is included in the MIS if and that it is excluded if (if , no constraint applies). We now prove that is independent, maximal, robust, and suitable in .

    Independence: Let be an edge of . If is an edge from one of the extra paths, and cannot be both in by construction. Otherwise (i.e., is in ), either belongs to , or it does not. If it does, then the clause introduced on Line 15 ensures that and cannot be both in the MIS. If it does not, then it belongs to , which is bipartite, then Line 08 (and the greedy process for extra paths) ensures that and cannot be both in the MIS.

    Maximality and robustness: Recall that independence is not affected by the removal of edges (Observation 1), thus robustness means preserving maximality. Let be a vertex of that does not belong to . Either belongs to an edge in or it does not. If it does, then by Remark 2 it has a neighbor in the MIS (maximality) and it cannot be a weak vertex (robustness). If it does not, then all of its neighbors in are in the set because is bipartite (and the extension of in the extra paths keep alternating the inclusion status), thus so long as remains connected, cannot be added to (maximality and robustness).

    Suitability: For every such that . Lines 12 and 13 ensure that only if PI appears in . Lines 10 and 11 ensure that only if PO or PE appear in . If and , then by construction and it is weak in . Line 08 implies that at least one adjacent edge to in is in and then PE does not appear in , implying that does.

  2. Let be a suitable RMIS of in which the status of is (resp ). Let . If is not bipartite, then it contains a weak vertex, which contradicts the robustness of . Thus, must be bipartite. In general, may be made of several connected components . A satisfying assignment can then be constructed from as follows. For all ranging from to , chose the value of so that . This assignment satisfies the mutual exclusivity constraint in Line 15 (with respect to edges in ), as otherwise would not be independent, and it also satisfies the constraint added with respect to (Lines 09 to 13), by suitability of . Finally, due to Lines 17 and 19, the assignment must include (respectively exclude) if the input flag is (resp. ). Thus a satisfiable assignment corresponding to must exists.

4.5.4.2  The Labeling Rule

Let us now return to the labeling of node in the -tree, assuming that all in are correctly labeled. The goal here is to label as per the possibilities, namely to assign label PI if an RMIS exists in that includes ; to assign label PO if an RMIS exists in that excludes ; and, in case the latter does not, to assign PE if an RMIS exists that excludes provided that an external neighbor of it is subsequently included.

We need to distinguish here between the case that is the root component of the -tree, or just an internal node. If is the root, then it has no attachment point and the only thing that matters is whether an RMIS exists or not. As such, does not receive a labeling, and is instead treated directly from the main algorithm (described in the next section). Thus, we focus on how the above procedure can be used to label when is an internal node of the -tree.

The first two tests are realized by fixing the status of as a parameter when calling . A first call is made, setting this parameter to in, then a second call is made, setting this parameter to . If the second call fails (no such RMIS exist), then a third call is made with a different strategy. This strategy relies on a trick using label PO. Remark 2 implies that whenever a child of is labeled PO, then (itself in ) does not need to have a neighbor within that is included in the RMIS. As a result, testing whether admits an RMIS if an external neighbor of is subsequently added (i.e., label PE) can be done by pretending that the parent of , whose attachment vertex is in , itself has an external neighbor in the MIS. Thus, just like the other children labeled PO, an artificial PO label is assigned to , resulting in the forced non-selection of to the MIS and the adjunction of an extra path to