Decomposition of Map Graphs with Applications

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a √(k)×√(k)-grid as a minor, or its treewidth is O(√(k)). However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs. Informally, our lemma states the following. For any map graph G, there exists a collection (U_1,...,U_t) of cliques of G with the following property: G either contains a √(k)×√(k)-grid as a minor, or it admits a tree decomposition where every bag is the union of O(√(k)) of the cliques in the above collection. The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time 2^O(√(k)k)· n^O(1) for the Connected Planar F-Deletion problem (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions. For Longest Cycle/Path, these are the first subexponential-time parameterized algorithms on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known 2^O(k^0.75k)· n^O(1)-time algorithms on map graphs.

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1 Introduction

In this paper, we develop new proof techniques to design parameterized subexponential-time algorithms for problems on map graphs, particularly problems that involve hitting or connectivity constraints. The class of map graphs was introduced by Chen, Grigni, and Papadimitriou [10, 11] as a modification of the class of planar graphs. Roughly speaking, map graphs are graphs whose vertices represent countries in a map, where two countries are considered adjacent if and only if their boundaries have at least one point in common; this common point can be a single common point rather than necessarily an edge as standard planarity requires. Formally, a map is a pair defined as follows (see Figure 1): is a plane graph111That is, a planar graph with a drawing in the plane. where each connected component of is biconnected, and is a function that maps each face of to or . A face of is called nation if and lake otherwise. The graph associated with is the simple graph where consists of the nations of , and contains for every pair of faces and that are adjacent (that is, share at least one vertex). Accordingly, a graph is called a map graph if there exists a map such that is the graph associated with .

Every planar graph is a map graph [10, 11], but the converse does not hold true. Moreover, map graphs can have cliques of any size and thus they can be “highly non-planar”. These two properties of map graphs can be contrasted with those of -minor free graphs and unit disk graphs: the class of -minor free graphs generalizes the class of planar graphs, but can only have cliques of constant size (where the constant depends on ), while the class of unit disk graphs does not generalize the class of planar graphs, but can have cliques of any size. At least in this sense, map graphs offer the best of both worlds. Nevertheless, this comes at the cost of substantial difficulties in the design of efficient algorithms on them.

Arguably, the two most natural and central algorithmic questions concerning map graphs are as follows. First, we would like to efficiently recognize map graphs, that is, determine whether a given graph is a map graph. In 1998, Thorup [33] announced the existence of a polynomial-time algorithm for map graph recognition. Although this algorithm is complicated and its running time is about , where is the number of vertices of the input graph, no improvement has yet been found; the existence of a simpler or faster algorithm for map graph recognition has so far remained an important open question in the area (see, e.g., [12]).

The second algorithmic question—or rather family of algorithmic questions—concerns the design of efficient algorithms for various optimization problems on map graphs. Most well-known problems that are NP-complete on general graphs remain NP-complete when restricted to planar (and hence on map) graphs. Nevertheless, a large number of these problems can be solved faster or “better” when restricted to planar graphs. For example, nowadays we know of many problems that are APX-hard on general graphs, but which admit polynomial time approximation schemes (PTASes) or even efficient PTASes (EPTASes) on planar graphs (see, e.g., [4, 17, 18, 25]). Similarly, many parameterized problems that on general graphs cannot be solved in time unless the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [28] fails, admit parameterized algorithms with running times subexponential in on planar graphs (see, e.g., [1, 2, 17, 31]). It is compelling to ask whether the algorithmic results and techniques for planar graphs can be extended to map graphs.

For approximation algorithms, Chen [9] and Demaine et al. [15] developed PTASes for the Maximum Independent Set and Minimum -Dominating Set problems on map graphs. Moreover, Fomin et al. [24, 25] developed an EPTAS for Treewidth- Modulator for any fixed constant , which encompasses Feedback Vertex Set and Vertex Cover. For parameterized subexponential-time algorithms on map graphs, the situation is less explored. While on planar graphs there are general algorithmic methods—in particular, the powerful theory of bidimensionality [18, 16]—to design parameterized subexponential-time algorithms, we are not aware of any general algorithmic method that can be easily adapted to map graphs. Demaine et al. [15] gave a parameterized algorithm for Dominating Set, and more generally for -Center, with running time on map graphs. Moreover, Fomin et al. [24, 25] gave -time parameterized algorithms for Feedback Vertex Set and Cycle Packing on map graphs. Additionally, Fomin et al. [24, 25] noted that the same approach yields -time parameterized algorithms for Vertex Cover and Connected Vertex Cover on map graphs. However, the existence of a parameterized subexponential-time algorithm for Longest Path/Cycle222In the Longest Path/Cycle problem, we ask whether a given graph contains a path/cycle on at least vertices. Here, the parameter is . on map graphs was left open. Furthermore, time complexities of , although having subexponential dependency on , remain far from time complexities of and that commonly arise for planar graphs [31]. We remark that time complexities of and are particularly important since they are often known to be essentially optimal under the aforementioned ETH [31].

In the field of Parameterized Complexity, Longest Path/Cycle , Feedback Vertex Set and Cycle Packing serve as testbeds for development of fundamental algorithmic techniques such as color-coding [3], methods based on polynomial identity testing [29, 30, 34, 6], cut-and-count [14], and methods based on matroids [22]. We refer to [13] for an extensive overview of the literature on parameterized algorithms for these three problems on general graphs. By combining the bidimensionality theory of Demaine et al. [16] with efficient algorithms on graphs of bounded treewidth [20, 13], Longest Path/Cycle, Cycle Packing and Feedback Vertex Set are solvable in time on planar graphs. Furthermore, the parameterized subexponential-time “tractability” of these problems can be extended to graphs excluding some fixed graph as a minor [18].

Our results and methods

Our results. We design parameterized subexponential-time algorithms with running time for a number of natural and well-studied problems on map graphs.

Let be a family of connected graphs that contains at least one planar graph. Then Connected Planar -Deletion (or just -Deletion) is defined as follows.

-Deletion Input: A graph and a non-negative integer . Question: Is there a set of at most vertices such that does not contain any of the graphs in as a minor?

-Deletion is a general problem and several problems such as Vertex Cover, Feedback Vertex Set, Treewidth- Vertex Deletion, Pathwidth- Vertex Deletion, Treedepth- Vertex Deletion, Diamond Hitting Set and Outerplanar Vertex Deletion are its special cases. We give the first parameterized subexponential algorithm for this problem on map graphs, which runs in time . Our approach for -Deletion also directly extends to yield -time parameterized algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on map graphs. (In this versions we are asked if there is a connected vertex cover or a feedback vertex set of size at most .)

With additional ideas, we derive the first subexponential-time parameterized algorithm on map graphs for Longest Path/Cycle. Our technique also allows to improve the running time for Cycle Packing (does a map graph contains at least vertex-disjoint cycles) from to .

Our results are summarized in Table 1.

Methods. The starting point of our study is the technique of bidimensionality [18, 16]. The core engine behind this technique is a combinatorial lemma of Robertson, Seymour and Thomas [32] that states that every planar graph either has a -grid as a minor, or its treewidth is . Unfortunately, a clique on vertices has no -grid as a minor and its treewidth is . Because classes of geometric graphs such as unit disk graphs and map graphs can have arbitrarily large cliques, the combinatorial lemma is inapplicable to them. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Specifically, every unit disk graph has a natural partition of such that each part induces a clique with “nice” properties—in particular, it has neighbors only in a constant number (to be precise, this constant is at most 24) of other parts; it was shown that either has a -grid as a minor, or it has a tree decomposition where every bag is the union of of these cliques [23]. In particular, given a parameterized problem where any two cliques have constant-sized “interaction” in a solution, it is implied that any bag has -sized “interaction” with all other bags in a solution. For any map graph , there also exists a natural collection of subsets of that induce cliques with “nice” properties. However, not only are these cliques not vertex disjoint, but each of these cliques can have neighbors in arbitrarily many other cliques.

In this paper, we first prove that every map graph either has a -grid as a minor, or it has a tree decomposition where every bag is the union of of the cliques in the above collection. For -Deletion, Connected Vertex Cover, and Connected Feedback Vertex Set, this combinatorial lemma alone already suffices to design -time algorithms on map graphs. Indeed, we can choose a fixed constant so that in case we have a -grid as a minor, there does not exist a solution, and otherwise we can solve the problem by using dynamic programming over the given tree decomposition. Specifically, since every bag is the union of cliques, and the size of each clique is upper bounded by (once we know that no -grid exists), only vertices in the bag are not to be taken into a solution—there are only choices to select these vertices, and once they are selected, the information stored about the remaining vertices is the same as in normal dynamic programming over a tree decomposition of width.

This approach already substantially improves upon the previously best known algorithms for Feedback Vertex Set, Vertex Cover and Connected Vertex Cover of Fomin et al. [24, 25]. However, -time algorithms for Longest Path/Cycle and Cycle Packing on map graphs require more efforts. The main reason why we cannot apply the same arguments as for unit disk graphs is the following. Recall that for unit disk graphs, given a parameterized problem where any two cliques have constant-sized “interaction” in a solution (in our case, this means a path/cycle on at least vertices, or a cycle packing of cycles), it is implied that any bag has -sized “interaction” with all other bags in a solution. Here, interaction between two cliques refers to the number of edges in a solution “passing” between these two cliques; similarly, interaction between a bag and a collection of other bags refers to the number of edges in a solution that have one endpoint in and the other endpoint in some bag in the collection. In this context, dealing with map graphs is substantially more difficult than dealing with unit disk graphs. In map graphs vertices in a clique can have neighbors in arbitrarily many other cliques in the collection rather than only in a constant number as in unit disk graphs. This is why it is difficult to obtain an -sized “interaction” as before.

This is the reason why we are forced to take a different approach for map graphs by bounding “the interaction within a clique across all the bags of a decomposition”. Towards this, we first need to strengthen our tree decomposition. To explain the new properties required, we note that every clique in the aforementioned collection of cliques, say , is either a single vertex or the neighborhood of some “special vertex” in an exterior bipartite graph (see Section 2). Further, every vertex of occurs as a singleton in . We construct our decomposition in a way such that every bag is not necessarily a union of cliques in , but a union of carefully chosen subcliques of cliques in (with one subclique for each of these cliques); subcliques of the same clique chosen in different bags may be different. We then prove properties that roughly state that, if we look at the collection of bags that include some vertex of , then this collection induces a subtree and a path as follows: the subtree consists of the bags that correspond to the singleton clique , and the path goes “upwards” (in the tree decomposition) from the root of this subtree. We thereby implicitly derive that in every bag , every subclique of size larger than can only have as neighbors vertices that are (i) in the bag itself or in one of its descendants, or (ii) in cliques that have a subclique in the bag . In particular, this means that if we prove that there exists a solution such that for any clique in , the number of edges in that “cross any bag ” (i.e., the edges in with one endpoint in and the other in the collection of all bags that are not descendants of ) is a constant, then we obtain a bound of on the interaction between any bag and the collection of all bags that are not descendants of . We prove the mentioned statement using property . The proof that such a property simultaneously holds for all cliques and all bags is the most challenging part of the proof.

We remark that we discussed above two types of tree decompositions, first the special one and then its stronger variant which is used for Longest Path/Cycle and Cycle Packing. Since the stronger variant of the decomposition can be used to work with -Deletion too, in the technical part of this paper we derive only the stronger variant of the decomposition. In Section 2, we give definitions, notations and some known results which we use throughout the paper. In Section 3, we design a tree decomposition of map graphs which we call as few clqiues tree decomposition, and in Section 4 we explain its direct applicability for Feedback Vertex Set and -Deletion. In Sections 5 and 6 we design subexponential-time parameterized algorithms for Longest Path/Cycle and Cycle Packing on map graphs, respectively. For these two problems we need additional, somehow technically involved, combinatorial “sublinear crossing” lemmata.

2 Preliminaries

The set of natural numbers is denoted by . For any , we use and as shorthands for and , respectively. For a set , we use to denote the power set of . Two disjoint sets and , we use to denote the disjoint union of and . For a sequence and any , the sequence is called a segment of . For a sequence and a subset , the restriction of on , denoted by , is the sequence obtained from by deleting the elements of .

Standard graph notations.

We use standard notation and terminology from the book of Diestel [19] for graph-related terms that are not explicitly defined here. Given a graph , let and denote its vertex-set and edge-set, respectively. When the graph is clear from context, we denote and . For a set of graphs we slightly abuse terminology and let and denote the union of the sets of vertices and edges of the graphs in , respectively. A graph is simple if it contains neither loops nor multiple edges between pairs of vertices. Throughout the paper, when we use the term graph we refer to a simple graph. Given , let denotes the subgraph of induced by . For an edge subset , let denotes the set of endpoints of the edges in , and denotes the graph with vertex set and edge set . Given , let denotes the edge set . Moreover, let denotes the open neighborhood of in ; we omit the subscript when the graph is clear from context. In case , we slightly abuse terminology and use . For a graph and a vertex , let . A graph is called a minor of if can be obtained from by a sequence of edge deletions, edge contractions, and vertex deletions. For a graph and a degree- vertex , by contracting , we mean deleting from and adding an edge between the two neighbors of in .

In a graph , a sequence of vertices is a path in if for any distinct , , and for any , . We also call the path as - path, and its internal vertices are . For any two paths and with , let denotes the path . A sequence of vertices is a cycle in if , is a path, and . Since in a multi graph there can be more than one edges between a pair of vertices, we use sequence to denote a cycle. In that context, for each , is an edge between and . For a graph , we say that is a clique if is a complete graph. Given , an grid is a graph on vertices, for , such that for all and , it holds that and are neighbors, and for all and , it holds that and are neighbors.

A binary tree is a rooted tree where each node has at most two children. In a labelled binary tree, for each node with two children one of the children is labelled as “left child” and the other child is labelled as “right child”. A postorder transversal of a labelled binary tree is the sequence of where for each node , appears after all its descendants, and if has two children, then the nodes in the subtree rooted at the left child appear before the nodes in the subtree rooted at the right child. For a binary tree , we say that a sequence of is a postorder transversal if there is a labelling of such that is its postorder transversal.

A tree decomposition of a graph , which is defined as follows, measures how close the graph is to a tree like structure.

[Treewidth] A tree decomposition of a graph is a pair , where is a rooted tree and is a function from to , that satisfies the following three conditions. (We use the term nodes to refer to the vertices of .)

• .

• For every edge , there exists such that .

• For every vertex , the set of nodes induces a (connected) subtree of .

The width of is . Each set is called a bag. Moreover, denotes the union of the bags of and its descendants. The treewidth of is the minimum width among all possible tree decompositions of , and it is denoted by .

A nice tree decomposition is a tree decomposition of a form that simplifies the design of dynamic programming (DP) algorithms. Formally,

A tree decomposition of a graph is nice if for the root of , it holds that , and each node is of one of the following types.

• Leaf: is a leaf in and . This bag is labelled with leaf.

• Forget vertex: has exactly one child , and there exists a vertex such that . This bag is labelled with forget.

• Introduce vertex: has exactly one child , and there exists a vertex such that . This bag is labelled with introduce.

• Join: has exactly two children, and , and . This bag is labelled with join.

We will use the following two folklore observations in Section 3. The correctness of these observations follows from Condition of a tree decomposition.

Observation .

Let be a nice tree decomposition of a graph . For any , there is exactly one node such that is labelled with forget.

Observation .

Let be a nice tree decomposition of a graph , , and be the node labelled with forget. For any node in the subtree of rooted at and , either or .

The following proposition concerns the computation of a nice tree decomposition.

[[7]] Given a graph and a tree decomposition of , a nice tree decomposition of the same width as can be computed in linear time.

Planar Graphs and Map Graphs.

A graph is planar if there is a mapping of every vertex of to a point on the Euclidean plane, and of every edge of to a curve on the Euclidean plane where the extreme points of the curve are the points mapped to the endpoints of , and all curves are disjoint except on their extreme points.

[Theorem 7.23 in [13],[27, 32]] For any , every planar graph of treewidth at least contains a grid minor. Furthermore, for every , there exists an time algorithm that given an -vertex planar graph and , either outputs a tree decomposition of of width at most , or constructs a grid minor in .

By substituting in Lemma 2, we get the following corollary.

There exists an time algorithm that given an -vertex planar graph and , either outputs a tree decomposition of of width less than , or constructs a grid minor in .

Map graphs are the intersection graphs of finitely many connected and interior-disjoint regions of the Euclidean plane. Any number of regions can meet at a common corner which results (in the map graph) in a clique on the vertices corresponding to these regions. Map graphs can be represented as the half-squares of planar bipartite graphs. For a bipartite graph with bipartition , the half-square of is the graph with vertex set and edge set is defined as follows: two vertices in are adjacent in if they are at distance in . It is known that the half-square of a planar bipartite graph is a map graph [10, 11]. Moreover, for any map graph , there exists a planar bipartite graph such that is a half-square of  [10, 11]; we refer to such as a planar bipartite graph corresponding to the map graph (see Figure 2).

Throughout this paper, we assume that any input map graph is given with a corresponding planar bipartite graph  333This assumption is made without loss of generality in the sense that if is given with an embedding instead to witness that it is a map graph, then is easily computable in linear time [10, 11].. We remark that we consider map graphs as simple graphs, that is, there are no multiple edges between two vertices and , even if there are two or more internally vertex disjoint paths of length between and in the corresponding planar bipartite graph. For a map graph with a corresponding planar bipartite graph having bipartition , we refer to the vertices in simply as vertices and the vertices in as special vertices. Moreover, we denote the special vertices by . Notice that for any , forms a clique in ; we refer to these cliques as special cliques of . We remark that the collection of cliques mentioned in Section 1 refers to .

3 Few Cliques Tree Decomposition of Map Graphs

In this section, we define a special tree decomposition for map graphs. This decomposition will be derived from a tree decomposition of the bipartite planar graph corresponding to the given map graph. Once we have defined our new decomposition, we will gather a few of its structural properties. These properties will be useful in designing fast subexponential time algorithms on map graphs.

Let be a map graph with a corresponding planar bipartite graph . Let be a tree decomposition of of width less than . A pair is called the -few cliques tree decomposition derived from , or simply an -FewCliTD, if it is constructed as follows (see Figure 3).

1. The tree is equal to . Whenever and are clear from context, we denote both and by .

2. For each node , . That is, for each node , we derive from by replacing every special vertex by .

In words, the second item states that for every vertex and node , we have that if and only if either or for some and for some node in the subtree of rooted at .

Next, we prove that the -FewCliTD in Definition 3 is a tree decomposition of . We remark that if we replace the term by the term in the second item of Definition 3, then we still derive a tree decomposition, but then some of the properties proved later do not hold true.

Let be a map graph with a corresponding planar bipartite graph . Let be an -FewCliTD where is a tree decomposition of of width less than . Then, is a tree decomposition of .

Proof.

We first prove that every vertex of is present in at least one bag. Towards this, notice that Property of Definition 2 of the tree decomposition of implies that . Therefore, since for any , we conclude that . Now, we prove that for any edge , there exists a bag for some such that . Because , there exists a special vertex such that . By Property of , the set of nodes induces a (connected) subtree of . Let be a node such that the distance from to the root of is minimized. Therefore, the choice of is unique. Since , by Property of and the definition of , there exist such that and . Then, because and must be descendants of in , we have that . This implies that . So we have proved Properties and of Definition 2.

To prove Property of Definition 2 with respect to , we pick an arbitrary vertex , and prove that the set of nodes induces a (connected) subtree of . Observe that where , and for each . To prove that is connected, it is enough to prove that is connected, for all , and is connected for all . Statement follows from Property of the tree decomposition of . For any , since and by Property of , we have that , and hence Statement follows.

The proof of the lemma will be complete with the proof of Statement . Towards this, let for all . Clearly . By Property of , we know that for any , induces a (connected) subtree of . We claim that for any , (which is a subgraph of ) is a (connected) subtree of . Towards a contradiction, suppose that is not connected. Then, let and be two connected components of such that there exists a path in from a vertex in to a vertex in whose internal vertices all belong to . Then, there is an internal vertex of such that is an ancestor of one of the end-vertices of (because in a rooted tree any internal vertex of a path is an ancestor of an end-vertex of the path and has at least one internal vertex, else and form one connected component). This implies that , because and is an ancestor a vertex in (where by the definition of , must belong to the bag of that vertex). This is a contradiction to the fact that . Therefore, we conclude that is connected. This completes the proof of the lemma. ∎

To simplify statements ahead, from now on, we have the following notation.

Throughout the section, we fix a map graph , a corresponding planar bipartite graph of , an integer , a nice tree decomposition of of width less than and an -few cliques tree decomposition of derived from using Definition 3

Recall that and that for each node , was obtained from by replacing every special vertex with .

For a node , we use Original() to denote the set , Fake to denote the set , and Cliques to denote the set of special cliques of . Informally, for a node , Original() denotes the set of vertices of present in the bag , Fake denotes the set of “new” vertices added to while replacing special vertices in , and Cliques is the set of special cliques in that consist of one for each special vertex . For example, let be the node in Figure 3 that is labelled with forget() by . Then, Original, Fake and Cliques.

In the remainder of this section we prove properties related to and , which we use later in the paper to design some of our subexponential-time parameterized algorithms. Towards the formulation of the first property, consider the tree decomposition in Figure 3 and the set of its nodes whose bags contain the vertex as a “fake” vertex. This set of nodes forms a path with one end-vertex being the unique node of labelled with forget by and the other end-vertex being an ancestor of . In fact, the set of nodes forms the unique path in from to where is the unique child of the node labelled with forget by . This observation is abstracted and formalized in the following lemma.

Let and such that and . Let be the node in labelled with forget by , and be the unique child of the node labelled with forget by . Then, is an ancestor of , and induces a path in which is the unique path between and in .

Proof.

First, we prove that induces a (connected) subtree of . Suppose not. Then, there exist two connected components and of such that there exists a path in from a vertex in to a vertex in whose internal vertices all belong to . By Property of the tree decomposition , we have that for any . Moreover, there is an internal vertex of such that is an ancestor of one of the end-vertices of . This implies that , because belong to the bags of the endpoints of (by the definition of and ). As we have also shown that for all , this implies that , which is a contradiction. Hence, we have proved that is connected.

Next, we prove that is a path such that one of its endpoints is a descendant of the other. Towards this, it is enough to prove that for any distinct , either is a descendant of or is a descendant of . For the sake of contradiction, assume that there exist such that neither is a descendent of nor is a descendent of . By the definition of and because , we have that and . Thus by Property of the tree decomposition , we have that and . Because and , this is a contradiction to the definition of .

It remains to prove that is an ancestor of and that and are endpoints of . Towards this, recall that is the node in labelled with forget by the tree decomposition . First, we prove that is an end-vertex of the path . Let be the only child of . To prove is an end-vertex of the path , it is enough to show that and . Since is labelled with forget by , we have that , , and . This implies that and hence . Now, we prove that . For this purpose, let . Clearly, . By Property of the tree decomposition , we have that is connected. We have already proved that is a path and since , is a path in . Since is labelled with forget by , for any node in the subtree rooted at and , by Observation 2, either or . This implies that contains no node in the subtree of rooted at and not equal to . Moreover, observe that there exists a node in the subtree of rooted at such that and hence . Now, since is non-empty and is connected, we have that . Since , and , we conclude that . Thus, we have proved that is an end-vertex of the the path .

Next we prove that is the other end-vertex of the path and is an ancestor of . Since is the only child of the node labelled with forget, we have that and . This implies that . Thus to prove that is an end-vertex of the path , it is enough to prove that . Since , , and is the parent of , by Property of , we have that is an ancestor of . This also implies that and . Hence, . This completes the proof of the lemma. ∎

In the next lemma we show that for any special vertex and any node in labelled with introduce by , it holds that and its child carry the “same information”.

Let and be a node in labelled with introduce by . Let be the only child of . Then, and .

Proof.

We know that . This implies that . To prove that , it is enough to show that (because any special vertex in and not equal to , is also belongs to ). Suppose by way of contradiction that and let . Then, there is a descendent of such that