Vertex-Connectivity Measures for Node Failure Identification in Boolean Network Tomography

12/04/2018 ∙ by Nicola Galesi, et al. ∙ 0

We investigate three questions in Boolean Network Tomography, related to maximal vertex identifiability, i.e. the maximal number of failing nodes simultaneously identifiable in a network. First, how to characterize the identifiability of the network through structural measures of its topology; second, how many monitors and where to place them to maximize identifiability of failures; third, which tradeoffs there are between the number of monitors and the maximal number of identifiable failures. We first consider Line-of-Sight networks and we characterize the maximal identifiability of such networks highlighting that vertex-connectivity plays a central role. Motivated by this observation, we give a precise characterization of the maximal identifiability in terms of vertex-connectivity for any network: using Menger's theorem, we prove that maximal identifiability is contained in a constant of 1 from vertex connectivity. A consequence of this is a first algorithm based on the well-known Max-Flow problem to decide where to place the monitors in any network in order to maximize identifiability. Finally we initiate the study of maximal identifiability for random networks on two models: Erdös-Rènyi model and Random Regular graphs. The framework proposed in the paper allows a probabilistic analysis of the identifiability in random networks giving a tradeoff between the number of monitors to place and the maximal identifiability.

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1 Introduction

A central issue in the study of communication networks is that of constantly ensuring that the structure works reliably. To this end it is of the utmost importance to discover as quickly as possible those components which are no longer working properly and present some sort of failure. As hinted by its name, Network Tomography is a family of distributed failure detection algorithms based on the spreading of end-to-end measurements [Var96, CHNY02]. Research in Network Tomography is vast. Methods and algorithms vary dramatically depending on the type of failure of interest of the measurements one has to rely on. Boolean Network Tomography, (BNT for short), is a family of methods to identify corrupted components in a network using boolean measurements routed through end-to-end paths. Introduced in [Duf06, GKAT11], BNT has recently attracted a lot of interest given the importance for example of recognizing failing nodes or links in a network, and given the simple measurements ( failing/ working) we runs through the network. In this work we use BNT to identify corrupted/failing nodes. We investigate three general questions.

Q1.

How to characterize the maximal identifiability of the given network through structural measures of the network topology?

Q2.

How to place monitors and how many of them we use to maximize the identifiability of failures?

Q3.

What is the tradeoff between the number of monitors and the maximal number of identifiable failures?

Assume to have a set of measurement paths over a node set . We would like to know the state (with corresponding to “ in working order” and corresponding to “ in a faulty state”) of each node . The localization of the failing nodes in is captured by the solutions of the system:

(1)

where models the (boolean) state of the path . Of course, systems of this form may have several solutions and therefore, in general, the availability of a collection of end-to-end measurements does not necessarily lead us to the unique identification of the failing nodes. We will investigate properties of the underlying network that facilitate the solution of this problem.

Finally note that, when measurements and variable states are allowed to take arbitrary real values, the expression (1) generalizes to the linear system , where is a matrix, whose entry is an indicator for the event “”;

is a vector of

real variables and . In this work we consider only the restricted boolean case, but our arguments can be extended to linear systems over .

1.1 Maximal Failure Identifiability

In studying the connection between the space of solution of systems like (1) and the identifiability of failing nodes in a network by means of end-to-end measurements along paths in we follow the approach initiated by L. Ma, T. He et. al. in [MHS14] based on the notion of maximal identifiability, (see Section 2.5 for a precise formal definition of such concept). The metric aims to capture the maximal number of failing nodes that can be simultaneously identified in a network described by the path system . It turns out that is an interesting combinatorial measure and in fact in several recent works [BHH17, GR18, MHS14, RD16] it has been analyzed also in connection with various properties in graphs. Here we follow [MHS14] and relate to the host graph vertex-connectivity.

1.2 Previous Works

Designing a network with high identifiability or finding heuristics to place monitors in order to maximize the identifiability of failure of various types is clearly an important problem

[Duf06]. Recently [RD16] studied such properties when looking at edge failure identifiability. In this work the maximal indentifiability was studied in connection with the host graph edge connectivity. In [MHL14, HGM17] L. Ma et al. gave necessary and sufficient conditions on the network topology for uniquely identifying link metrics from controllable, cycle-free measurements between the monitors.

Relatively less is known about node failure identifiability [MHS14, MHS17, BHH17]. Galesi and Ranjbar [GR18] gave tight upper and lower bounds for for several classes of specific network topologies, such as hypergrids, using as structural measure of the network its minimal degree and upper bounding with for any graph .

1.3 Summary of Contributions

In an attempt to generalize the results of [GR18] on hypergrids, we start by studying the identifiability of Line-of-Sight (LoS for short) networks. LoS networks were introduced by Frieze et al. in [FKRD07] and have been widely studied (see for instance [DF13, CW07, SZ17]) as they can be used to model communication patterns in a geometric environment containing obstacles. Like grids, LoS networks can be embedded in a finite cube of , for some positive integer . But LoS networks generalize grids in that edges are allowed between nodes that are not necessarily next to each other in the network embedding (see subsection 2.3 and Figure 2). The first important contribution of our work is a complete characterization of the maximal identifiability of Line-of Sight networks. It turns out that the approach suggested [GR18], based on minimal degree, works for LoS networks too, but in fact it can be improved. Using the network vertex-connectivity, , (i.e. the size of the minimal set of node disconnecting the graph) rather than its minimum degree, we are able to prove that

(2)

for any LoS network .

It is well-known in graph theory (see Lemma 2.1) that . Thus the result on LoS networks immediately suggests the related question about general graphs. In this work we devise a general approach to node failure identifiability based on vertex-connectivity, considering the question of understanding the precise relationship between vertex-connectivity and maximal identifiability for general networks. We prove upper and lower bounds for in terms of . Precisely we have that

for any graph having (see Corollary 4.2 and 4.6 below). The upper bound (which in fact is valid for any graph ) is proved by a technique similar to the one used in [GR18] when working with . The lower bound, which requires to find paths separating big sets of nodes in the graph, is instead proved by using a tool from Graph theory: Menger’s Theorem (see Theorem 4.4 below).

In the last section we initiate the study of maximal identifiability for random networks by looking at Erdős-Rényi and Random Regular Graphs. We show a trade-off between the success probability of the various random processes to reach a maximal identifiability and the size of the sets

and . Random graphs also give us constructions of networks with large identifiability.

1.4 Organization of the Paper

After a preliminary section, where we give all the important definitions used in the paper including that of maximal identifiability, we have a Section dedicated to the results on Line-of-Sight graphs. Section 4 describes the connectivity bounds on . Section 5 is dedicated to the analysis of maximal identifiability on random graphs. It is divided in three subsections. First we analyze the case of Erdős-Rényi graphs and we show a simple analysis to prove sublinear maximal identifiability. A more refined analysis is sketched in the second subsection for Erdős-Rényi graphs reaching an optimal linear separability. Finally we analyse maximal identifiability for the case of random regular graphs. All our results give immediate tradeoffs between maximal identifiability and number of monitors.

2 Preliminaries

2.1 Sets, Graphs, Paths

If and are sets, is the symmetric difference between and . In a graph , is a set of nodes and is the collection of (undirected) edges. We assume has no loops of parallel edges. A path (of length ) in from a node to a node is a sequence of nodes such that , and for all . Any sub-sequence (, ) of is said to be contained in . We say that path and intersect if there is at least one sequence of nodes that is contained in both paths. Sometimes we also talk about the intersection of a path and an arbitrary set of nodes . Such expression denotes the set of elements of that are contained in . For a node in , is the set of neighbourhood of , i.e. . The degree of , , is the cardinality of . In what follows , the minimum degree of , and , the maximum degree of .

2.2 Connectivity

In what follows (resp. ) will denote the vertex-connectivity (resp. edge-connectivity) of the given graph . Namely is the size of the minimal subset of , such that removing the vertices in from disconnect , and is the size of the minimal set such that removing all edges in from disconnect .

The following inequalities are well-known (see for example [Har69], Theorem 5.1, pag 43).

Lemma 2.1.

For any graph , .

It will also be convenient to work with sets of vertices disconnecting particular parts of the given graph. If , then is the size of the smallest vertex separator of and in , i.e. the smallest set of vertices whose removal disconnects and (set if ). Notice that .

2.3 Grids and LoS Networks

For positive integers , and , let be the -dimensional cube . We say that distinct points and in one of these cubes share a line of sight if their coordinates differ in a single place. A graph is said to be a Line of Sight (LoS) network of size , dimension , and range parameter if there exists an embedding such that if and only if and share a line of sight and the (Manhattan) distance between and is less than . In the rest of the paper a LoS network is always given along with some embedding in for some and , and with slight abus de langage we will often refer to the vertices of , in terms of their corresponding points in , and in fact the embedding will not be mentioned explicitly.

Definition 2.2 (Hypergrids).

Let and , . The hypergrid of dimension over support , , is the graph with vertex set and where here is an edge between a node and a node if for some we have and for all .

In the case of simple grids over nodes, i.e. , we use the notation . For , the border is the set of border nodes such that for some .

Note that -dimensional hypergrids as defined in [GR18] are particular LoS networks with . In the forthcoming sections we will also study augmented hypergrids (or simply in the 2-dimensional case), namely -dimensional LoS networks with range parameter containing all possible nodes.

2.4 Models

In this paper we work with undirected graphs, but we assume that measurements originate at a certain collection of source monitors, and are collected at a collection of target monitors. Each measurement signal originates a particular source, follows a path in , and ends in one of the target monitors. In fact monitors have a special status, and we may safely assume that they are not part of the network. However, to highlight the possibility that some nodes in might be directly connected to path monitors, we denote graphs as tuples where is the nodes directly linked to input monitors, is the set of nodes directly linked to output monitors, and is the set of all other internal nodes (as usual is the set of edges). We do not assume any direction in the edges of in the sense that, for different choices of and , each particular edge might be traversed in either directions.

Figure 1: Small graph with , .

The ability to identify node failures depend on the system of paths we use to take our measures. In this work we assume to be able to probe the vertices of the given along any simple path connecting the given and (this is related to model CSP in [MHS14]). We call any such path and - path. In what follows, denote the set of all simple paths in starting at a node in , terminating at a node of and passing through , and , for any set . In general we omit the subscript when the set and is clear from the context. In the graph in Figure 1 and are examples of simple paths in . Notice in particular that, if one traverses the paths from to edge is traversed starting at and moving to in , and starting at (moving to ) in .

2.5 Identifiability

The main goal of BNT in the context of this paper is to devise algorithms that allow us to detect node failures in networks and also identify which nodes resulted in such failures. Such identification, essentially, boils down to studying the solutions of systems like (1) but, as pointed out in Section 1, often there is many of them and the identification can be tricky. To address this problem Ma et al. [MHS14] suggested that it might be useful to exploit the structure of the host graph.

The notion of identifiability, introduced in [MHS14] and developed in numerous subsequent papers [MHL14, MHS17, BHH17, GR18] within the context of both node and link failure studies, plays a central role. Following [GR18], we formulate identifiability for a set of paths in terms of set operations. Let be a set of end-to-end paths over nodes .

Definition 2.3.

Let . A graph . is -identifiable iff for all , with and , it holds that .

Notice that if is -identifiable, for any set of at most nodes in , there is at least one path that uniquely identify , against another set . Therefore if, in a given set of measurements, fails, we can identify the failure of , rather than . Also, identifiability obeys a notion of monotonicity: if is -identifiable, then is -identifiable, for any positive integer . Hence of particular importance is the value of the large for which is -identifiable.

Definition 2.4 (Maximal identifiability).

Let be a graph. The maximal identifiability of , is the largest integer for which is -identifiable.

Notice that the previous definition depends on and . Maximizing (G) over all possible pairs, we can define . Differently than we have that for any .

3 Failure Identifiability in LoS Networks

Let be an integer. In this section we analyze the maximal identifiability of LoS networks starting with , and later considering the general case. The following Lemma states a simple fact about whichs will help us to prove a upper bounds on using Lemma IV.4 in [GR18].

Lemma 3.1.

Let , and . .

Proof.

In , each node has edges for each one of the possible directions (north, south, east, west) the node is linked to. In an internal node these are , in border nodes and in corner nodes . Hence the minimal degree in is reached at the corner nodes and it is . ∎

By Lemma IV.4 of [GR18] we have . In the remainder of this section we pair this up with a tight lower bound. Note that has many more edges than the simple grid (studied in [GR18]). Theoerm 3.2 below shows that four monitors are, in general, not enough. In general, we place input monitors on the west and north borders of and output monitors on the south and east borders of . Given a node of , identified as a pair , we define:

and

(when talking about LoS network we will often identify a node in the given graph with the point it is mapped to). We are now ready to state the main result of this section.

Theorem 3.2.

Let , and . Assume that has input monitors and output placed as described above. Then .

Proof.

We have to prove that given two node sets , and of cardinality at most , with we can build an - path touching exactly one of them. Given a node , let , the nodes in the North-West region of and let , the nodes in South-East region of . Notice that and and . Since and , there is a node in . Assume that (if is similar and give even better results). Similarly for , assume that . Consider the following definition:

Definition 3.3.

Given a node , a set of nodes in . We say that a direction (north, south, west, east) is -saturated on if moving from on direction there is, right after , a consecutive block of nodes in .

The following claim define two disjoint paths in from to and in form to not touching . Their concatenation hence defines a - path passing from and not touching and proves the theorem. ∎

Claim 3.4.

Let . There is a path in from to not touching . There is a path in from to not touching .

Proof.

We prove the first one since they are the same. By induction on . If , then and we have done. If . Since , and since a direction is -saturated only if a block of consecutive elements of appear after on that direction, then there is at a least a direction between North and West which is not -saturated. Hence there is a node on direction from at distance less than . Hence there is an edge . Since the inductive hypothesis give us a path as required. Hence the path is as required. ∎

Theorem 3.2 generalizes to the case of LoS networks in dimensions.

3.1 General Line-of-Sight Networks and Vertex-Connectivity

Figure 2: Example of a LoS () embedded in (here represented as a dashed grid).

Assume now to have an arbitrary Line-of-Sight network embedded in (see Figure 2 for an example). First let us define a set of the rules to place the monitors in which are somehow inherited from the approach used in .

Definition 3.5.

(Corner monitors) Let be an arbitrary 2-dimensional LoS network. Define:

i.e. a node is (linked to) an input monitor, if and only if it is the only node in the NW region of in ; and

i.e. a node is (linked to) an output monitor, if and only if is the only node in the SE region of in .

We contend that the minimum degree does not provide the best way to describe the network maximal identifiability in LoS graphs. The example in Figure 2 has minimum degree three but, we claim, maximal identifiability only one. Vertex-connectivity seems to be important when a vertex cut separates input from output monitors. A proof that maximal identifiability can be upper bounded by minimal vertex-connectivity is in Theorem 4.1 in the next section for general graph. Here we prove that for generic LoS graphs the maximal identifiability can be also lower bounded by vertex-connectivity. The proof follows a similar idea of previous theorem.

Theorem 3.6.

Let be an arbitrary 2-dimensional LoS network such that , where is the size of the minimal vertex cut separating from . Then .

Proof.

(Sketch) Let us fix and of cardinality such that . Take a in . Define the following set of nodes in : and Notice that and similarly . Let (resp. ) be the node in (resp. ) which exist by cardinality constraints. As in previous theorem 3.2 we build two disjoint paths (without cycles) from to and from to avoiding . We show how to build . is similar. By induction on . If then and the result follows.

Claim 3.7.

and .

Proof.

Let us prove the first one. Assume by contradiction that . Since and , then removing the nodes in from would separate from . This could not happen since the size of the minimal vertex-cut in separating from is . ∎

By previous claim and the fact that , it follows that there is a node . , hence by induction there is a path from to not touching . Define . ∎

4 Vertex Connectivity, Identifiability and Optimal Monitor Placement

In the work [GR18] the authors characterized precisely in terms of but only for specific class of topologies, like trees and hypegrids. Since (see Lemma 2.1) we raise the question of what can be said about relationships between and . For general graphs we completely characterize in terms of establishing a precise, graph theoretical and new vision of the maximal identifiability measure. To understand the relationship between identifiability, minimal degree, and vertex connectivity interacts each other, consider the graph in Figure 3 (A). Notice that but, we argue, . Let be the edge disconnecting . Set and . Notice that each and all paths from to necessarily touch both and . Hence , using previous Lemma.

(A)

(B)

Figure 3: Picture (A): a small graph with and identifiability 0 (here is the set of dark vertices on the left-hand side and the set on dark squares right-hand side). Picture (B): a different arrangement of the monitors in the same graph changes its identifiability.

A key observation is that the (edge) cut separates the input from output monitors. If input and output monitors are not separated by , as in Figure 3 , we cannot conclude that But only use the bound , which is still 5.

We formalise precisely this property to improve the upper bound on minimal degree of [GR18]. Let . Let be the size of the minimal vertex set separating form in .

In the graph in Figure 3 (A) and are -vertex separable.

Theorem 4.1.

Let be a graph. Then .

Proof.

Let be the set witnessing the minimal separability of from in . Hence . Let be the set of nodes neighbours of nodes in and notice this cannot be empty since is disconnecting . Pick one and define and . Clearly . To see the opposite inclusion assume that there exists a path from to passing from but not touching . Then is not separating from in . Contradiction. ∎

Recall form subsections 2.2,2.5 that for all , and that . Since previous theorem holds in particular for the pair witnessing , i.e. , then we have.

Corollary 4.2.

Let be a graph. Then .

4.1 Lower Bounds

We have seen how to upper bound in terms of the smallest separator of the given monitor sets. What can be said about lower bounds? Lower bounds are much more interesting since they require to build paths connecting the two monitor sets in the graph avoiding certain sets of nodes. In this section we prove that can always be lower bounded in terms of the host graph vertex connectivity. Our main result is the following:

Theorem 4.3.

Let be a graph and assume . Then .

The result is inspired by the approach used to deal with LoS networks, but in this general case we need to work with the given and . The key points in Theorem 3.6 are:

  1. for any node , we are guaranteed to have an acyclic path connecting to and touching ; and

  2. the paths and in , respectively connecting to and to , are disjoint since they connect nodes in disjoint regions of the underlaying grid: and the .

For general graphs, P1 will require a more elaborate construction. To capture property P2 in graphs where it is not possible to define the regions and the , we use the following well known result (see for instance [Har69, Theorem 5.10, p. 48]):

Theorem 4.4 (Menger’s Theorem).

A graph is -connected if and only if each pair of nodes is connected by node-disjoint paths.

(two paths between two nodes and are node-disjoint if they have no common nodes except for and ). It is worth to notice that in fact we will use Menger’s Theorem twice: once to “defeat” minimal connectivity and a second time to bypass cyclicity between and .

To prove that is identifiable we need to argue that for any given and any one can find a path from some to some going through but not touching any node in . The following result is instrumental to the proof of such property.

Lemma 4.5.

Let be an arbitrary graph, and let be a set of vertices in such that . Any pair of vertices in is connected by at least two vertex disjoint simple paths in .

Proof.

Assume by contradiction that for some pair there is at most one simple path between and in . Each simple path, in , between and , except the one in , must use exactly one element of , and each of these paths must use different elements of . Hence there are vertex disjoint paths connecting and in , and this (via Menger’s Theorem) contradicts the assumption that the smallest separator between and has size . ∎

Proof (of Theorem 4.3) Let and be such that . We will show that . Let , and be two subsets of of cardinality (the result for smaller sets is easier). We want to find a path connecting to in touching only one between and . Take and assume wlog that . Since is connected there is a simple path starting at some , ending at some and passing through . If does not touch we are done. Call the segment of that starts in and ends at and the rest of . Assume that the intersection of and is contained in . Notice that since and , there is a node . Notice that if . By Lemma 4.5 applied on , we have two vertex-disjoint paths and connecting to and not touching at all. We use these two paths to build a path from to touching but avoiding . More precisely, if at least one of and does not intersect (say path ), we are done: It suffices to set , the concatenation of and . We hence assume that both and touch in one or more nodes. Say these sets of nodes are respectively and . Define an order on the nodes of as follows: if going from to we pass though . Let and be the smallest nodes respectively in and under . Assume wlog the (the other case is symmetrical). The path is defined from the concatenation of the following paths which are clearly disjoint:

  1. , a path contained in ;

  2. , a sub-path of ;

  3. the path , connecting to .

To complete the proof note that the case in which the intersection between and is contained in is identical to the previous one with and swapped (i.e. we use Lemma 4.5 to connect to ).


As for the upper bound, since an , Theorem 4.3 implies:

Corollary 4.6.

Let with . Then .

Proof.

The stated inequality holds if as applying Theorem 4.3 when and are separated by a minimum vertex cut requires

5 Random Network Models and Tradeoffs

The main results in this work extend our understanding of the relationship between a graph maximum identifiability and its vertex connectivity. In general to separate arbitrary sets of vertices and one needs repeated applications of Max-Flow. In this section we prove that simpler methods exist in at least two types of random graphs. Also we show an interesting trade-off between the success probability of the various random processes and the size of the sets and . Finally, random graphs give us constructions of networks with large identifiability.

5.1 Sub-Linear Separability in Erdős-Rényi Graphs

We start our investigation of the identifiability of node failures in random graphs by looking at the binomial model , for fixed . The following equalities, which hold w.h.p., are folklore:

(3)

(see [Bol01]). Here we describe a simple method which can be used to separate sets of vertices of sublinear size.

In what follows, the set of nodes attached to monitors, , is formed by two disjoint parts: the set , consisting of those nodes that are directly connected to input monitors, and , the set of nodes adjacent to output monitors. We assume, for now, that each of these parts is formed by nodes with nodes.

Figure 4: A node and a possible way to connect it to and .

Let and be two arbitrary subsets of of size . The probability that and are separable is at least the probability that an element of (w.l.o.g. assume ) is directly connected to a node in and to a node in . This event has probability Hence the probability that and cannot be separated is at most and therefore the probability that some pair of sets and of size (not intersecting ) fail is at most

Theorem 5.1.

For fixed with , under the assumptions above about the way monitors are placed in , the probability that is not -vertex separable is at most

Proof.

The argument above works if both and contain no vertex in . The presence of elements of vertices in in or may affect the analysis in two ways. First could be in (say ). In this case and are separable if is directly connected to a vertex in . This happens with probability Second, might contain some elements of and different from . In the worst case when is trying to connect to , it must avoid at most element of such set. There is at most pairs of and of size at most . Thus the probability that fails to be -vertex separable is at most and the result follows as . ∎

5.2 Linear Separability in Erdős-Rényi Graphs

The argument above cannot be push all the way up to . When trying to separate vertex sets containing vertices the problem is that these sets can form a large part of and the existence of direct links from to and is not guaranteed with sufficiently high probability. However a different argument allow us to prove the following:

Theorem 5.2.

For fixed , w.h.p.

Full details of the proof are left to the final version of this paper, but here is an informal explanation. The upper bound follows immediately from (3) and Theorem 4.1. For the lower bound we claim that the chance that two sets of size at most are not vertex separable is small. To see this pick two sets and , and remove, say, . is still a random graph on at least vertices and constant edge probability. Results in [BFF87] imply that has a Hamilton path starting at some and ending at some with probability at least (and in fact there is a fast polynomial time algorithm that finds one). Such Hamilton path, by definition, contains a path from to passing through , for every possible choice of . This proves, w.h.p., the separability of sets of size up to . Past such value the construction in Theorem 4.1 applies.

5.3 Random Regular Graphs

A standard way to model random graphs with fixed vertex degrees is Bollobas’ configuration model [Bol80]. There’s buckets, each with free points. A random pairing of these free points has a constant probability of not containing any pair containing two points from the same bucket or two pairs containing points from just two buckets. These configurations are in one-to-one correspondence with -regular -vertex simple graphs. Denote by the set of all configurations on buckets each containing points, and let -reg be a random -regular graph.

When studying vertex separability in random regular graphs, as for , we assume that vertices in a set are connected to sender monitors, and the same number in a set to terminal monitors (therefore there’s “internal” buckets). These define the set . The main result of this section is the following:

Theorem 5.3.

Let be a fixed integer. -reg w.h.p.

The upper bound follows immediately from the aforementioned connectivity result and Theorem 4.1. The lower bound is a consequence of the following:

Lemma 5.4.

Let be a fixed integer and a fixed positive number. Two sets and with -reg and are separable w.h.p. if .

If we know that the construction in Theorem 4.1 applies and therefore there exist -regular graphs which contain pairs of sets of size that are not separable. If the construction is not possible and Lemma 5.4 implies the lower bound in Theorem 5.3.

Proof.

of Lemma 5.4. We choose two sets of buckets, and . For simplicity assume that both and are subsets of . The probability that and can be separated is at least the probability that a (say) random element of (w.l.o.g. ) is connected to by a path of length at most and to by a path of length at most , neither of which “touch” .

Figure 5: Assume . The picture above represents a bucket (i.e. vertex) and two possible paths (length 4 and 5, respectively) connecting it to and .

Figure 5 provides a simple example of the event under consideration. The desired paths can be found as follows. First, starting from , build (say by performing a random walk) a simple path of length that avoids . We call RW such process. Similarly, again starting from , build a simple path of length that avoids . Either of these processes may fail if at any point we re-visit a previously visited bucket or if we hit or even .

Claim 5.5.

RW and RW succeed w.h.p. provided .

Any point in is useful. (resp. ) contain (resp ) useful points. The second part of the process finds a neighbour for all useful points in and . The process succeeds if we manage to hit one of the senders from and one of the receivers from . We call this process RandomShooting.

Claim 5.6.

RandomShooting succeeds w.h.p. if .

Proof.

(Sketch) A single useful point “hits” its target set, say , with probability proportional to the cardinality of . Hence the probability that none of the useful points hits is and the overall success probability is

After the simplication obtained by setting (and then using for ), the argument above implies that the success probability for and is asymptotically approximately and the rest of the argument (and its conclusion) is very similar to the case (the final bound is slightly weaker, though). The chance that a random -regular graph is not -vertex separable is at most

which goes to zero as provided is constrained as in the Lemma statement. ∎

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