A new lower bound for eternal vertex cover number

10/11/2019 ∙ by Jasine Babu, et al. ∙ The Fleet Street 0

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph G, in terms of the cardinality of a vertex cover of minimum size in G containing all its cut vertices. The consequences of the lower bound includes a quadratic time algorithm for computing the eternal vertex cover number of chordal graphs.

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1 Introduction

Eternal vertex cover number of a graph is the minimum number of guards required to successfully keep defending attacks on a graph, in a certain multi-round attack-defense game [1]. The rules to play the game with guards on a graph are the following. Initially, the defender places the guards on a subset of vertices of . The positions of the guards defines an initial configuration. In each round of the game, the attacker chooses an edge of to attack. In response to the attack, the defender is free to move each of guard from its current position to an adjacent vertex or retain it in its current position. All guards are assumed to move in parallel, at the same time. The constraint to be satisfied is that at least one guard should move from an endpoint of to the other. If the defender is able to successfully move the guards satisfying this constraint, we say that the attack in the current round is successfully defended. The resultant positions of the guards define the configuration from where the next round of the attack-defense game continues. If the defender can keep on successfully defending any sequence of attacks, we say that the defender has a defense strategy on this graph, with guards. Eternal vertex cover number of a graph , denoted by is the minimum integer such that the defender has a defense strategy on , with guards. When this game is played with -guards, each configuration encountered in the game is equivalent to some function from to such that (where, for each , will be the number of guards on ). A set of such configurations , such that the defender can start with any configuration in as the initial configuration and keep moving between configurations in for defending the attacks, is called an eternal vertex cover class of and each configuration in is an eternal vertex cover configuration. If is an eternal vertex cover class of such that the number of guards in the configurations in is equal to , then is a minimum eternal vertex cover class of . There are two popular versions of the game: the former in which in any configuration, at most one guard is allowed on a vertex and the latter in which this restriction is not there. Since the main structural result in this paper is a lower bound for eternal vertex cover number, we will be assuming the first version of the game. It can be easily verified that our proofs work the same way in the other model of the game as well.

From the description of the game, it is clear that, in any configuration, if at least one of the endpoints of an edge is not occupied, the defender will not be able to successfully defend an attack on that edge. Therefore, . This is the only general lower bound known for the parameter, so far in literature. In this work, we prove that the size of a minimum sized vertex cover of that contains all cut vertices of is also a lower bound for . This improved lower bound has many algorithmic consequences, including a quadratic time algorithm for computing the eternal vertex cover number of chordal graphs and a PTAS for computing the eternal vertex cover number of internally triangulated planar graphs. These results generalize the results presented in [2].

2 A new lower bound

Definition 1

Let be a graph and . The smallest integer , such that has a vertex cover of cardinality with , is denoted by .

Definition 2 (-components and -extensions)

Let be a cut vertex in a graph and be a component of . Let be the induced subgraph of on the vertex set . Then is called an -component of and is called an -extension of .

To simplify the expressions that appear later, we introduce the following notation. For any graph and any set , the notation will be used to denote the set .

Definition 3

Let be the set of cut vertices of a graph and let . The set of -components of will be denoted as . If is any block of , then the set of -components of is defined as
.

Definition 4 (EVC-Cut-Property)

Let be a graph and let be the set of cut vertices of . The graph is said to have the EVC-cut-property if for every graph that is an -extension of for some , it is true that in each eternal vertex cover configuration of , at least guards are present on the vertices of , out of which at least guards are present on .

Note 1

For a graph to satisfy the EVC-Cut-Property, it is not necessary that the vertex is occupied by a guard in every eternal vertex cover configuration of an -extension of . All the (or more) guards could be on vertices other than .

Note 2

Definition 4 gives some lower bounds on the number of guards and not on the number of vertices with guards. Note that, if more than than one guard is allowed on a vertex, then these two numbers could be different.

The following two lemmas are easy to obtain, using a straightforward counting argument.

Lemma 1

Let be a graph and be the set of cut vertices of . For any ,

Lemma 2

Let be a graph and be the set of cut vertices of . If is a block of and is any vertex of such that , then

Lemma 3

Every graph satisfies EVC-cut-property.

Proof

The proof is by induction on the number of blocks of the graph. First consider a graph with a single block. Let be any vertex of and be an -extension of . Let be an eternal vertex cover configuration of and let be the set of vertices of on which guards are present in . Since is an eternal vertex cover configuration of , must be a vertex cover of and must be a vertex cover of . Therefore, . If , then there are at least guards on and at least guards on , as we need to prove. Also, it is easy to see that . Therefore, we are left with the case when . This implies that . Thus, in the remaining case to be handled, the number of vertices on which guards are present is exactly and there is no guard on .

From this point, let us focus on the number of guards on and not just the number of vertices that are occupied. If there are more than guards in , then the conditions we need to prove are satisfied for the configuration . In the remaining case, we have exactly guards in , with . In this case, we will derive a contradiction.

Consider an attack on an edge incident at , where . Let be the new configuration, after defending this attack and be the set of vertices on which guards are present in . In the transition from to , a guard must have moved from to . Also, being a cut vertex, no guard can move from to . Therefore, . But, this is a contradiction because is a minimum vertex cover of containing , but we have .

Thus, the lemma holds for all graphs with only one block. Now, as induction hypothesis, assume that the lemma holds for any graph with at most blocks. We need to show that the lemma holds for any graph with blocks.

Let be an arbitrary graph with blocks and let be an arbitrary vertex of . Let be the set of cut vertices of and let be an arbitrary -extension of . Let be an arbitrary eternal vertex cover configuration of and let be the set of vertices on which guards are present in . Let . We need to show that there are at least guards on in and at least guards on . Let be the number of guards on in . We split our proof into two cases based on whether is a cut vertex in or not.

Case 1. is a cut vertex of :

In this case, by our induction hypothesis, for each -component of , at least guards are on in the configuration . There are two possible sub-cases.

  1. If is not occupied by a guard in , then by induction hypothesis,
    . Since is non-empty, by Lemma 1, it follows that . The number of guards on is .

  2. If is occupied by a guard in , still, in order to satisfy the induction hypothesis for all -components of , the number of guards on must be at least . Therefore, by Lemma 1, it follows that the number of guards on is at least and .

Case 2. is not a cut vertex of :

Let be the block of that contains . By Lemma 2, we have:

(1)

Before proceeding with the proof, we establish the following claim.

Claim

Suppose is an eternal vertex cover configuration of . Then the number of guards on in configuration is at least .

Proof

To count the number of guards on , we count the total number of guards on the -components of and the number of guards on the remaining vertices separately and add them up.

  • First, we will count the total number of guards on the -components of . For each -component of , let . For each cut vertex , let denote the family of -components of that intersect at the cut vertex and let denote . Consider a -component of . By our induction hypothesis, the number of guards on is at least in . Moreover, since is connected to by a single cut vertex, from the induction hypothesis it follows that the number of guards on is at least . Note that, for each cut vertex , the total number of guards on must be at least , to satisfy the above requirement. By summing this over all cut vertices in , the total number of guards on must be at least .

  • Now, we will count the number of guards on the remaining vertices. To cover the edges inside the block that are not incident at any vertex in , at least vertices of are to be occupied in . If is occupied in , then at least vertices of are occupied in . In either of these cases, the number of guards on is at least .

Therefore, the total number of guards on is at least . Since , we can conclude that the number of guards on is equal to . Comparing this expression with Equation 1, we can see that the number of guards on is at least . ∎

Now, we continue with the proof of Lemma 3. There are two possible sub-cases.

  1. If is occupied by a guard in , then by Claim 2, it follows that the number of guards on is at least and the number of guards on is at least , as we require.

  2. If is not occupied in , then by Claim 2, . If , we are done. If , then we will derive a contradiction. Consider an attack on an edge such that . While defending this attack, a guard must move from to . Let be the new configuration in and let be the set of vertices on which guards are present in . Note that no guards from can move to any vertex of in this transition from to , because is a cut vertex in . Therefore, in , the total number of guards on is less than , contradicting Claim 2. Therefore, and the lemma holds for .

Thus, by induction, the lemma holds for every graph. ∎

Remark 1

Note that, the above lemma holds for both the models of the eternal vertex cover; the first model in which the number of guards permitted on a vertex in any configuration is limited to one and the second model, where this restriction is not there. However, it is possible that, in the second model, the number of vertices on which guards are present could be smaller than . An example for this is shown in Figure 1.

Figure 1: Any vertex cover of the graph in (a) that contains vertex and both the cut vertices must be of size at least . The graph in (b) is a -extension of the graph in (a). Positions of guards in an eternal vertex cover configuration of the graph in (b) are indicated using gray squares. This is a valid configuration. Note that, only four vertices of the graph in (a) are occupied in the configuration shown in (b).
Theorem 2.1

For any connected graph , , where is the set of cut vertices of .

Proof

Let be an eternal vertex cover configuration of and be the set of all vertices of containing guards in . Suppose . Then, there exists a vertex such that . Since every graph satisfies EVC-cut-property by Lemma 3, for each -component of , exactly guards are present on . Therefore, the total number of guards is at least . Since there are at least two -components, by comparing this expression with the RHS of the equation in Lemma 1, we can see that the total number of guards is more than . This contradicts our initial assumption. ∎

Observation 1

Let be a connected graph and let be the set of cut vertices of . If , then in every minimum eternal vertex cover configuration of , there are guards on each vertex of .

Proof

For contradiction, assume that there exists a minimum eternal vertex cover configuration of with a cut vertex unoccupied. Rest of the proof is exactly the same as in the proof of Theorem 2.1. ∎

For any graph and , let denote the minimum number such that has an eternal vertex cover class with guards in which all vertices of are occupied in every configuration of . By Observation 1, we have the following generalization of Corollary 2 of [3].

Theorem 2.2

Let be a connected graph with at least two vertices and let be the set of cut vertices of . Suppose that every vertex cover of of size , such that , induces a connected subgraph in . If for every vertex , then, . Otherwise, .

Proof
  • By Theorem 2.1, we have and we have . If for every vertex , , then by Lemma 2 of [3], and hence, .

  • If for some vertex , , then by Theorem 1 of [3], . Let be any minimum sized vertex cover of that contains all vertices of . Since is a connected vertex cover of , by a result by Klostermeyer and Mynhardt [1], . Thus, we have .

A graph is locally connected if for every vertex of , its open neighborhood induces a connected subgraph in . If every block of a graph is locally connected, then every vertex cover of that contains all its cut vertices is connected. Hence, we have:

Corollary 1

Let be a connected graph with at least two vertices, such that each block of is locally connected and let be the set of cut vertices of . Then, . Further, if and only if for every vertex , .

The following remark is a generalization of Remark 3 of [2].

Remark 2

Let be a connected graph with at least two vertices and let be the set of cut vertices of . Suppose that for every vertex cover of of size such that , the induced subgraph is connected. Then, .

3 Algorithmic implications

In this section, we provide generalizations of some algorithmic results in [2], using results obtained in the previous section.

We define to be the graph class that consists of connected graphs such that for each , it is true that for every vertex cover of such that and , where is the set of all cut vertices of , the subgraph is connected.

The following result is a generalization of Corollary 3 of [2].

Observation 2

Given a graph and an integer , deciding whether is in NP.

Proof

Consider any with at least two vertices and let be the set of cut vertices of .
By Remark 2, . To check if , the polynomial time verifiable certificate consists of at most vertex covers of size at most such that for each vertex , there exists a vertex cover in the certificate containing all vertices of . ∎

3.1 Hereditary graph classes

The following theorem is obtained by generalizing Corollary 6 of [2], by applying Theorem 2.2.

Theorem 3.1

Let be a hereditary graph class such that each biconnected graph in is locally connected. If the vertex cover number of any graph in can be computed in time, then the eternal vertex cover number of any graph can be computed in time.

Proof

Let be a graph in . Since each block of is locally connected, by Corollary 1, . Further, by Corollary 1, to check whether , it is enough to decide if for every vertex , . Minimum vertex cover computation can be done for graphs of in time, for a vertex , checking whether , takes only time. Therefore, checking whether can be done in time. ∎

3.2 Chordal graphs

The following theorem is a special case of Theorem 3.1, using the fact that minimum vertex cover computation can be done for chordal graphs in time [4], where is the number of edges and is the number of vertices of the input graph. This result is a generalization of a result for biconnected chordal graphs in [2].

Theorem 3.2

Let be a chordal graph and be the set of cut vertices of . Then, and the value of can be determined in time, where is the number of edges and is the number of vertices of the input graph.

3.3 Internally triangulated planar graphs

The following lemma is a generalization of a result in [2] for biconnected internally triangulated planar graphs.

Lemma 4

Given an internally triangulated planar graph and an integer , deciding whether is in NP.

Proof

Since each block of an internally triangulated planar graph is locally connected, every vertex cover of that contains all its cut vertices induces a connected subgraph. Therefore, by Observation 2, deciding whether is in NP. ∎

The existence of a polynomial time approximation scheme for computing the eternal vertex cover number of biconnected internally triangulated planar graphs, given in [2], is generalized by the following result.

Lemma 5

There exists a polynomial time approximation scheme for computing the eternal vertex cover number of internally triangulated planar graphs.

Proof

Let be an internally triangulated planar graph. Let be the set of cut vertices of . It is possible to compute in linear time, using a well-known depth first search based method. By Remark 2, . It is easy to see that for a vertex , . Using the PTAS designed by Baker et al. [5] for computing the vertex cover number of planar graphs, given any , it is possible to approximate within a factor, in polynomial time. Therefore, there exists a polynomial time approximation scheme for computing . ∎

References

  • [1] Klostermeyer, W., Mynhardt, C.: Edge protection in graphs. Australasian Journal of Combinatorics 45 (2009) 235 – 250
  • [2] Babu, J., Chandran, L.S., Francis, M., Prabhakaran, V., Rajendraprasad, D., Warrier, J.N.: On graphs whose eternal vertex cover number and vertex cover number coincide (2019)
  • [3] Babu, J., Chandran, L.S., Francis, M., Prabhakaran, V., Rajendraprasad, D., Warrier, J.N.: On graphs with minimal eternal vertex cover number. In: Conference on Algorithms and Discrete Applied Mathematics (CALDAM), Springer (2019) 263–273
  • [4] Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM Journal on computing 5(2) (1976) 266–283
  • [5] Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1) (January 1994) 153–180