Connected ( s, t ) -Vertex Separator Parameterized by Chordality

We investigate the complexity of ﬁnding a minimum connected ( s, t )- vertex separator (( s, t )-CVS) and present an interesting chordality di-chotomy: we show that ( s, t )-CVS is NP-complete on graphs of chordal-ity at least 5 and present a polynomial-time algorithm for ( s, t )-CVS on chordality 4 graphs. Further, we show that ( s, t )-CVS is unlikely to have δlog 2 − (cid:15) n -approximation algorithm, for any (cid:15) > 0 and for some δ > 0, unless NP has quasi-polynomial Las Vegas algorithms. On the positive-side of approximation, we present a (cid:100) c 2 (cid:101) -approximation algorithm for ( s, t )- CVS on graphs with chordality c ≥ 3. Finally, in the parameterized setting, we show that ( s, t )-CVS parameterized above the ( s, t )-vertex connectivity is W [2]-hard.


Introduction
The vertex or edge connectivity of a graph and the corresponding separators are of fundamental interest in Computer Science and Graph Theory.For a connected graph, a vertex separator is a subset of vertices whose removal disconnects the graph into two or more connected components and the vertex connectivity refers to the size of a minimum vertex separator.Many kinds of vertex separators, stable vertex separators [1], clique vertex separators [18], constrained vertex separators [13], and α-balanced separators [13] are of interest to the research community.As far as complexity results are concerned, finding a minimum vertex separator and a clique vertex separator are polynomial-time solvable, whereas, finding a stable vertex separator and other constrained separators reported in [13] are NP-hard.This shows that imposing an appropriate constraint on the wellstudied vertex separator problem makes the problem NP-hard.Interestingly, constrained vertex separators have received much attention in parameterized complexity as well [13,12].In particular, Marx et al. in [13] considered the parameterized complexity of constrained separators satisfying some hereditary properties, for example, clique separators and stable separators.It is shown in [13] that the above problems have an algorithm whose running time is f (k) • n O (1) , where k is the size of a constrained separator.Algorithms of this nature are popularly known as fixed-parameter tractable algorithms (FPT) with parameter as the solution size [15].Subsequently, in [14], Marx et al. looked at the computational problem of finding a minimum (s, t)-vertex separator ((s, t)-CVS) satisfying some non-hereditary property, like connectedness.Interestingly, in [14] it is shown that (s, t)-CVS is in FPT.When a computational problem is known to be NP-complete, it is natural to look at the complexity of the same in special graph classes such as chordal graphs, P 5 -free graphs, planar graphs, etc. Well known problems such as maximum clique, maximum independent set, and minimum vertex cover have polynomialtime algorithms restricted to chordal graphs which are NP-complete in general graphs.Recent breakthrough due to Lokshtanov et al. [10] reveals that maximum independent set problem in P 5 -free graphs is polynomial time.Essentially, classical problems which are known to be NP-complete in general graphs have nice polynomial-time algorithms when the input is restricted to graphs with forbidden subgraphs.Moreover, this line of research has received a significant attention in the past as it helps to identify the gap between the NP-Hardness and the polynomial-time solvable input instances.Having highlighted the importance of special graph classes, in this paper, we investigate the complexity of (s, t)-CVS in chordal graphs (graphs with no induced cycle of length at least 3) and its super classes.It is a well-known fact that in chordal graphs every minimal vertex separator is a clique [7].It is clear that (s, t)-CVS is trivially solvable in chordal graphs.It is now natural to study (s, t)-CVS on graphs of higher chordality.A graph is said to have chordality c (c ≥ 3), if it does not contain any induced cycle of length at least c + 1.To the best of our knowledge the complexity of (s, t)-CVS in graphs of higher chordality (henceforth, chordality 1.As mentioned in the introduction, on chordal graphs every minimal vertex separator is a clique and therefore the (s, t)-CVS is immediately guaranteed in chordal graphs.Further, finding a minimum (s, t)-CVS in chordal graphs is equivalent to finding a minimum vertex separator which is polynomial-time solvable [7].We show that deciding (s, t)-CVS is NPcomplete on graphs of chordality 5 and on chordality 4 graphs (s, t)-CVS is polynomial-time solvable.This result is due to a very interesting structural property of minimal vertex separators in chordality 4 graphs and it says that every minimal vertex separator S is either connected or there exist two vertices u and v such that both u and v have a neighbour to each connected component of S in G.

2.
As far as approximation algorithms are concerned, we present two results.We first present a c 2 -approximation algorithm for (s, t)-CVS on graphs with chordality c ≥ 3. We then establish an approximation preserving polynomial-time reduction from the Group Steiner Tree [9,6] to (s, t)-CVS.Consequently, it follows that there is no polynomial-time approximation algorithm with approximation factor δlog 2− n for some δ > 0 and for any > 0, unless NP has quasi-polynomial Las Vegas algorithms.
3. Our final result is from parameterized complexity theory.As mentioned before Marx et al. [14] have shown that (s, t)-CVS is in FPT with parameter as the size of the connected vertex separator.Since an important lower bound for (s, t)-CVS is the (s, t)-vertex connectivity itself.It is now natural to consider the following parameterization: the size of a (s, t)-CVS minus the (s, t)-vertex connectivity.This type of parameterization is known as the above guarantee parameterization [11,8].We show that (s, t)-CVS parameterized above the (s, t)-vertex connectivity is unlikely to be fixed-parameter tractable under the standard parameterized complexity assumption, and in the terminology of parameterized hardness theory, it is hard for the complexity class W [2] in the W -hierarchy.
Graph Preliminaries: Notation and definitions are as per [7,16].Let G = (V, E) be a connected undirected unweighted simple graph where V (G) is the set of vertices and E(G) is the set of edges.For S ⊂ V (G), G[S] denote the graph induced on the set S and G \ S is the induced graph on the vertex set

Roadmap:
In Section 2, we analyze the complexity of (s, t)-CVS on chordality c graphs and present our dichotomy result.We then present an approximation algorithm with approximation ratio as a function of chordality of the graph.In Section 3, we present a classical and an approximation hardness for (s, t)-CVS.
We conclude Section 3 by presenting a parameterized hardness for the above guarantee (s, t)-CVS.
2 Complexity of (s, t)-CVS on Chordality c graphs The objective of this section is to look at the complexity of (s, t)-CVS with chordality as the parameter.Towards this end, we show that (s, t)-CVS is NPcomplete on chordality 5 graphs and we present a polynomial-time algorithm for (s, t)-CVS on chordality 4 graphs.We conclude this section with a c 2approximation algorithm for (s, t)-CVS on graphs of chordality c ≥ 3.In our reduction, we choose Steiner tree problem as the candidate problem and it is defined as follows; JGAA, 19(1) 549-565 (2015) 553 Steiner tree problem: Instance: A graph G, a terminal set R ⊆ V (G), and an integer r Question: Is there a subtree in G that contains all of R with at most r edges.Theorem 1 (s, t)-CVS is NP-complete on chordality 5 graphs.Proof: (s, t)-CVS is in NP: Given an input instance (G, s, t, q) of (s, t)-CVS, the certificate on Yes instances is a set S ⊆ V (G) which is a connected (s, t)-vertex separator of cardinality at most q.Clearly, S can be verified in polynomial time by standard reachability algorithms [2].(s, t)-CVS is NP-hard: It is known from [17] that Steiner tree problem on split graphs is NP-complete and this can be reduced in polynomial time to (s, t)-CVS in chordality 5 graphs using the following construction.Note that any split graph G can be seen as a graph with is an independent set.Also, split graphs are a subclass of chordal graphs and hence have chordality 3. We map an instance (G, R, r) of Steiner tree problem on split graphs to the corresponding instance (G , s, t, q = r + 1) of (s, t)-CVS as follows: An example is illustrated in Figure 1.We now show that instances created by this transformation have chordality 5. i.e., in G , any cycle C of length at least 6 has a chord.Clearly, C must contain either s or t but not both.Let {s, u 1 , . . ., u p }, p ≥ 5 denote the ordering of vertices in C.
Therefore, we conclude that chordality of G is at most 5.We now show that (G, R, r) has a Steiner tree with at most r edges if and only if (G , s, t, q = r + 1) has a (s, t)-CVS of size at most r + 1.For only if claim, G has a Steiner tree T containing all vertices of R and at most r edges.By our construction of G , to disconnect s and t, we must remove the set N G (s) which is R, as there is an edge from each element of N G (s) to t.Since G has a Steiner tree T with at most r edges, implies that T has at most r + 1 vertices.Clearly, in G , T guarantees a (s, t)-CVS of size at most r + 1.For if claim, G has a (s, t)-CVS S with at most r + 1 vertices.Note that any spanning tree on at most r + 1 vertices has at most r edges.From our construction of G , it follows that N G (s) ⊆ S and the (s, t)-vertex connectivity is |N G (s)|.This implies that G has a Steiner tree with at most r edges containing R = N G (s) as the terminal set.Hence the claim.
Hence, this is a polynomial-time reduction.As a consequence, it follows that (s, t)-CVS in chordality 5 graphs is NP-hard.Thus, we conclude (s, t)-CVS in chordality 5 graphs is NP-complete.

(s, t)-CVS in Chordality 4 Graphs is Polynomial time
In this section, we present the other half of our dichotomy result which says that (s, t)-CVS in chordality 4 graphs is polynomial-time solvable.We now present a sequence of combinatorial results on the structure of minimal vertex separators in chordality 4 graphs, using which we show that (s, t)-CVS in chordality 4 graphs is polynomial-time solvable.
Theorem 2 Every minimal (s, t)-vertex separator S in a chordality 4 graph G satisfies one of the following properties: and V (X i ) denotes the vertex set of the component X i .In G \ S, there exists u in C s and there exists v in C t such that for all is connected in G as well.Otherwise, by the induction hypothesis, in G • e, there exists u and v with the desired property.In particular, V (X ) ∩ N G•e (u) and V (X ) ∩ N G•e (v) are non empty where X = (X \ {x, y}) ∪ {z xy } and X is the connected component in S containing x and y.Since X is obtained from X and {x, y} ∈ E(G), it follows that u and v are adjacent to X in G. Thus, both u and v have the desired property in G too.A snapshot is illustrated in Figure 2.  wu denote the subpath of P s xu on the vertex set {w = w 1 , . . ., w q = u}, q ≥ 2. Let i, 2 ≤ i ≤ q be the smallest integer such that, {z, w i } ∈ E(G).In this case, P s xwi {w i , z}P t xz form an induced cycle of length at least 5 in G where P s xwi denote the subpath of P s xu on the vertex set {x, w = w 1 , . . ., w i }, 2 ≤ i ≤ q.Note that {x, z} / ∈ E(G) as S is an independent set.However, this contradicts the fact that G is a graph of chordality 4. Therefore, there exists a vertex û ∈ {u, w} in C s with the desired property.i.e., either u or w is adjacent to each element (connected component) in S. The proof for the existence of vertex v in C t is symmetric.A snapshot is illustrated in Figure 3. Proof: Note that any minimum (s, t)-CVS is of size at least k as the (s, t)vertex connectivity is k.If a minimum (s, t)-vertex separator itself is connected then we get a minimum (s, t)-CVS of size k.Otherwise, every minimum (s, t)vertex separator S is such that G[S] is a collection of connected components.
In this case, we know from Theorem 2, there exists a vertex v in one of the components of G \ S such that v has a neighbour in each connected component of S. Therefore, S ∪ {v} is a minimum (s, t)-CVS of size k + 1.Hence, the lemma is true.
Remark: For a chordality 4 graph G with the (s, t)-vertex connectivity k, asking for a minimum (s, t)-CVS of size k is equivalent to checking whether G contains a connected minimum (s, t)-vertex separator, i.e. a minimum (s, t)-vertex separator which itself is connected.The Lemma 2 shows that this equivalence checking is indeed polynomial-time solvable.
We now present two more combinatorial observations using which we can find a minimum (s, t)-CVS in chordality 4 graphs in polynomial time.We make use of the notion of contractible edges.Proof: From Lemma 2, it is clear that checking for a connected minimum (s, t)-vertex separator in G is equivalent to checking whether G • F contains a cut-vertex or not.This testing can be done using Depth First Search tree computed on G • F and hence, the claim.
Lemma 3 For a chordality 4 graph G with the (s, t)-vertex connectivity k, deciding whether (s, t)-CVS is of size k or k + 1 is polynomial-time solvable.
Proof: The claim follows from Lemmas 1, 2 and Corollary 1.The decision algorithm DECIDE-(s,t)-CVS(G,k) performs the following two tasks, namely, contract all non-contractible edges in G and check the (s, t)-vertex connectivity in the resulting graph G .If κ(G ) ≥ 2, then the algorithm returns 'NO' which means that every minimum (s, t)-CVS is of size k + 1.Otherwise, it returns 'YES' which means that there exists a minimum (s, t)-CVS of size k.Since the above tasks can be done using the standard depth first search algorithm, the decision algorithm runs in polynomial time.

Finding a minimum (s, t)-CVS in Chordality 4 graphs
Using DECIDE-(s,t)-CVS(), we now show that finding a minimum (s, t)-CVS in chordality 4 graphs is also polynomial-time solvable.The approach is to contract all non-contractible edges (edges in the set F ) and check whether the resulting graph contains a cut-vertex or not.If there is no cut-vertex, then any minimum (s, t)-vertex separator in G together with the vertex v in one of the components in G \ S (due to Theorem 2) yields a (s, t)-CVS of size k + 1 in G. Otherwise, the given chordality 4 graph contains a (s, t)-CVS of size k.
In such a case, we outline a procedure using which we can find a (s, t)-CVS S of size k.Our procedure (Algorithm 1) makes polynomial number of calls to DECIDE-(s,t)-CVS() to output the desired set.Find a minimum (s, t)-vertex separator S in G using classical vertex connectivity algorithm

5:
Output the set S ∪ {v} where v is in one of the components of G \ S such that S ⊆ N G (v), is a minimum (s, t)-CVS 6: else 7: /*---there exists a k-size (s, t)-CVS.To obtain one such separator, perform the following; ---*/ /* continue goes to the beginning of the for-loop */ 7: return x 8: end for Proof: From Lemma 1, we know that a minimum (s, t)-CVS is of size k or k +1.To decide between k and k +1, it is sufficient to check for a cut-vertex in G•F as per Lemma 2. This step can be implemented in polynomial time, by identifying the edges which are elements of F .Every edge whose contraction reduces the connectivity by 1 is in F .Then G • F is checked for the presence of a cut-vertex, and this can be done by a DFS.If the size of the minimum (s, t)-CVS is k + 1, then steps 4 and 5 of Algorithm 1 outputs a (s, t)-CVS of size k+1 in polynomial time, by finding a minimum (s, t)-vertex separator.If the minimum (s, t)-CVS is of size k, then Algorithm 2 returns a minimum connected (s, t)-CVS.Overall, a minimum (s, t)-CVS can be obtained in polynomial time.

( c
2 )-Approximation for (s, t)-CVS on Graphs with Chordality c Lemma 6 Let G be a graph of chordality c ≥ 3.For each minimal vertex separator S, for each u, v ∈ S such that {u, v} / ∈ E(G), there exists a path of length at most c 2 whose internal vertices are in C s or C t , where C s and C t are components in G \ S containing s and t, respectively.
Proof: Suppose for some non-adjacent pair {u, v} ⊆ S, both P 1 uv and P 2 uv are of length more than l 2 , where P 1 uv and P 2 uv are shortest paths from u to v whose internal vertices are in C s and C t , respectively.Now, there is an induced cycle C containing u and v such that |C| > l 2 + l 2 = l.However, this contradicts the fact that G is of chordality l.
Let OP T denote the size of any minimum (s, t)-CVS on chordality c graphs.Clearly, OP T ≥ k, where k is the (s, t)-vertex connectivity.The description of approximation algorithm ALG is as follows: Theorem 4 Algorithm 3 outputs (s, t)-CVS in polynomial time with approximation ratio c 2 .Proof: Observe that S is a (s, t)-CVS in G.The upper bound on the size of S output by ALG is: . Therefore, approximation ratio β is Algorithm 3 Approximation Algorithm for (s, t)-CVS on Chordality c Graphs 1: Compute a minimum (s, t)-vertex separator S in G. S = {v 1 , . . ., v k } be an arbitrary ordering of vertices in S 2: for each non-adjacent pair find a path P vivi+1 of length at most c 2 whose internal vertices are in C s or C t .Such a path exists as per Lemma 6 4: Step 1 of the Algorithm 3 incurs O(n 3 ) time to output a minimum (s, t)-vertex separator in G.To implement step 3, we can make use of the standard reachability algorithm like Breadth First Search (BFS) to output P vivi+1 and this call is made for at most O(n 2 ) time.Therefore, the overall time-complexity of the Algorithm 3 is (mn 2 ), where O(m) is the time incurred for BFS subroutine.
3 Complexity of (s, t)-CVS: Hardness Results The purpose of this section is two fold.Although in [14] it is shown that (s, t)-CVS is FPT, no explicit reduction is shown to establish NP-hardness result.In this section, we first establish a classical hardness of (s, t)-CVS by presenting a polynomial-time reduction from the Group Steiner tree to (s, t)-CVS.Moreover, the same reduction establishes an hardness of approximation for (s, t)-CVS.We conclude this section by showing that (s, t)-CVS parameterizing above the (s, t)vertex connectivity is W [2]-hard.

Classical Hardness: A Reduction from Group Steiner tree to (s, t)-CVS
The decision version of (s, t)-CVS is given below Instance: A graph G, a non-adjacent pair (s, t), and q ∈ Z + Question: Is there a (s, t)-vertex separator S ⊂ V (G), |S| ≤ q and G[S] is connected?
The Group Steiner tree problem can be stated as follows: given a connected undirected unweighted graph G, an integer r, and a collection of sets, which we call groups g 1 , g 2 , . . ., g l ⊆ V (G), the objective is to find a subtree T of G with at most r edges that contains at least one vertex from each group g i .We assume that the groups are disjoint.The Group Steiner tree problem is a generalization of the Steiner tree problem [5] and therefore, it is NP-complete.
We transform an instance I = (G, g 1 , g 2 , . . ., g l ⊆ V (G), r) of the Group Steiner tree to the corresponding instance I = (G , s, t, l+r+1) of (s, t)-CVS as follows: An example is illustrated in Figure 4. Proof: To establish NP-hardness result, we prove the following claim.For I and I as defined above, G has a Group Steiner tree with at most r edges if and only if G has a (s, t)-CVS of size at most r + 1 + l.We first prove the necessity.Given that G has a Group Steiner tree T with at most r edges that contains at least one vertex from each group g i .By the construction of G , it is clear that the (s, t)-vertex connectivity is l.Therefore, any (s, t)-CVS in G has at least l vertices.Clearly, these l new vertices together with at most r + 1 vertices in T form a (s, t)-CVS of size at most r + 1 + l in G .Conversely, by the construction of G , any (s, t)-CVS S of size at most r + 1 + l must contain all x i 's.i.e.N G (s) ⊂ S.This is true because N G (s) is a (s, t)-vertex separator.Since S is connected and N G (s) is an independent set, it follows that by the construction S \ N G (s) is connected.Moreover, S must contain at least one element of N G (x i ) for each x i .Since |S \ N G (s)| ≤ r + 1, any spanning tree on S \ N G (s) is a Group Steiner tree with at most r edges.As a consequence of the above claim, it follows that (s, t)-CVS is NP-hard and it is easy to verify that (s, t)-CVS is in NP as certificate testing can be done in polynomial time using standard graph traversals [2].Therefore, (s, t)-CVS is NP-complete.
We now show that our reduction establishes a stronger result: (s, t)-CVS is unlikely to have δ log 2− n-approximation algorithm, for any > 0 and for some δ > 0, unless NP has quasi-polynomial Las Vegas algorithms.
Hardness of Approximation of (s, t)-CVS: The Group Steiner tree problem with l groups is at least as hard as the Set Cover problem, thus can not be approximated to a factor o(log l), unless P = N P [4].On the hardness of approximation due to [9], the following result is known: there is no polynomial-time approximation algorithm for Group Steiner tree with approximation factor δ log 2− n for some δ > 0 and for any > 0, unless NP has quasi-polynomial Las Vegas algorithms.We now show that the above reduction is an approximation-ratio preserving reduction.Let OP T g and OP T c denote the size of any optimum solution of the Group Steiner tree problem and the (s, t)-CVS problem, respectively.Note that OP T c = OP T g + l and OP T g ≥ l.
Suppose there is an (1 + α)-approximation algorithm for (s, t)-CVS, where α ≤ δ log 2− n, for some δ, > 0. Then the size of the output of the algorithm is This implies 2(1 + α)-approximation algorithm for the Group Steiner tree problem, which is unlikely, unless NP has quasi-polynomial Las Vegas algorithms [9].
3.2 (s, t)-CVS Parameterized above the (s, t)-vertex connectivity is W [2]-hard We consider the following parameterization which is the size of (s, t)-CVS minus the (s, t)-vertex connectivity.Since the size of every (s, t)-CVS is at least the (s, t)-vertex connectivity, it is natural to parameterize above the (s, t)-vertex connectivity and its parameterized version is defined below.
(s, t)-CVS parameterized above the (s, t)-vertex connectivity: Instance: A graph G, a non-adjacent pair (s, t) with the (s, t)-vertex connectivity k and r ∈ Z + Parameter: r Question: Is there a (s, t)-vertex separator S ⊂ V (G), |S| ≤ k + r such that G[S] is connected?
We now show that there is no fixed-parameter tractable algorithm for (s, t)-CVS parameterized above the (s, t)-vertex connectivity.In order to characterize those problems that do not seem to admit a fixed-parameter tractable algorithms, Downey and Fellows defined a parameterized reduction and a hierarchy of intractable parameterized problem classes above FPT, the popular classes are W [1] and W [2]. We refer [15] for details about parameterized reductions.We now present a parameterized reduction from parameterized Steiner tree problem to (s, t)-CVS parameterized above the (s, t)-vertex connectivity.This parameterized version of Steiner tree problem is shown to be W [2]-hard in [3].Clearly, the reduction is a parameter preserving parameterized reduction.Therefore, we conclude that deciding whether a graph has a (s, t)-CVS is W [2]-hard with parameter r.
Concluding Remarks and Further Research: In this paper, we have investigated the complexity of minimum connected (s, t)-vertex separator ((s, t)-CVS) on graphs of higher chordality as finding a minimum (s, t)-CVS in chordal graphs is polynomial-time solvable.We have presented a chordality dichotomy which says that (s, t)-CVS is NP-complete on chordality 5 graphs and polynomial-time solvable on chordality 4 graphs.Further, we have presented a c 2 -approximation algorithm on graphs with chordality c ≥ 3. We also reported a non-approximiability result and in the parameterized-setting, we have established that parameterizing above the (s, t)-vertex connectivity is W [2]-hard.An interesting problem for further research is to parameterize (s, t)-CVS by the (s, t)-vertex connectivity.

Figure 2 :
Figure 2: A snapshot illustrating Case 1 of Theorem 2

Figure 4 :
Figure 4: An instance of Group Steiner tree reduces to an instance of (s, t)-CVS

Parameterized
Steiner tree problem: Instance: A graph G, a terminal set R ⊆ V (G), and an integer r Parameter: r Question: Is there a set of vertices T ⊆ V (G) \ R such that |T | ≤ r and G[R ∪ T ] is connected?T is called Steiner set (Steiner vertices).

Theorem 6
(s, t)-CVS Parameterized above the (s, t)-vertex connectivity is W [2]-hard.Proof: Given an instance (G, R, r) of Steiner tree problem, we construct the corresponding instance (G , s, t, k, r) of (s, t)-CVS with the (s, t)-vertex connectivity k = |R| as follows:V (G ) = V (G)∪{s, t} and E(G ) = E(G)∪{{s, v} | v ∈ R} ∪ {{t, v} | v ∈ R}.We now show that (G, R, r) has a Steiner tree with at most r Steiner vertices if and only if (G , (s, t), k, r) has a (s, t)-CVS of size at most k + r.For only if claim, G has a Steiner tree T containing all vertices of R and at most r Steiner vertices.By our construction of G , to disconnect s and t, we must remove the set N G (s) which is R, as there is an edge from each element of N G (s) to t.Since G has a Steiner tree with at most r Steiner vertices, implies that in G , it guarantees a (s, t)-CVS of size at most k + r.For if claim, G has a (s, t)-CVS S with at most k + r vertices.Since the (s, t)-vertex connectivity is k and S is a (s, t)-vertex separator, from our construction of G it follows that N G (s) ⊆ S and k = |N G (s)|.This implies that G has a Steiner tree with R = N G (s) as the terminal set and S \ N G (s) as the Steiner vertices of size at most r.Hence the claim.|V (G )| = |V (G)| + 2 and |E(G )| ≤ |E(G)| + 2|V (G)| and the construction of G takes O(|E(G)|).
s and t are in two different connected components and S -vertex separator S, let C s and C t denote the connected components of G \ S such that s is in C s and t is in C t .We let G • e denote the graph obtained by contracting the edge e = {u, v} in G such that V (G • e) = V (G) \ {u, v} ∪ {z uv } and E(G • e) = {{z uv , x} | {u, x} or {v, x} ∈ E(G)} ∪ {{x, y} | {x, y} ∈ E(G) and x = u, y = v}.
Such a w exists as S is a minimal (s, t)-vertex separator in G.If for all z ∈ S, {w, z} ∈ E(G), then w is a desired vertex in C s .Otherwise, there exists z ∈ S such that {w, z} / ∈ E(G).Let P s G•xy, there exists u in C s and v in C t satisfying our claim where C s and C t are connected components in (G • xy) \ S containing s and t, respectively.Let S = {x, y, u 1 , . . ., u p }, p ≥ 0. We now prove in G the existence of vertex u in C s satisfying our claim.If {u, x}, {u, y} ∈ E(G), then clearly u ∈ C s is the desired vertex in G. Otherwise, without loss of generality assume that x / ∈ N G (u).Thus, S \ {x} ⊂ N G (u).Let P s xu denote a shortest path between x and u such that the internal vertices are in C s .Consider the vertex w in P s xu such that {x, w} ∈ E(G).
Given a connected graph G with the (s, t)-vertex connectivity k, an edge e ∈ E(G) is said to be contractible if the (s, t)-vertex connectivity in G • e is at least k.Otherwise e is called noncontractible.For a connected graph G with the (s, t)-vertex connectivity k ≥ 2, let F = {{u, v} | {u, v} ∈ E(G) and {u, v} is contained in a minimum (s, t)vertex separator }. i.e., the set F is the set of all non-contractible edges in G.We use F to denote the set of non-contractible edges in G.By G • F , we mean the graph obtained from G by contracting all edges in F .Suppose, G • F does not contain a cut-vertex.This implies that after contracting edges in F , in G • F , every minimum vertex separator S induces at least two connected components.Moreover, this is true even in G as well, contradicting the fact that there exists a connected minimum (s, t)-vertex separator in G.Only if: Suppose every minimum (s, t)-vertex separator S is such that G[S] has at least two connected components.Since any edge contraction can not disconnect a graph which is already connected, any sequence of edge contractions of edges in F results in a graph with the vertex connectivity at least two, contradicting the fact that G • F contains a cut-vertex.Hence, the claim follows.
Computing the set F : The set F can be computed in polynomial time.Given a graph G with the (s, t)-vertex connectivity k, for each edge e in G, compute G • e and check whether the (s, t)-vertex connectivity is k − 1.If so, then e ∈ F .Proof: If:Corollary 1 For a connected graph G, deciding whether G contains a connected minimum (s, t)-vertex separator is polynomial-time solvable.
Lemma 4Let G be a chordality 4 graph with the (s, t)-vertex connectivity k ≥ 2. G has a (s, t)-CVS of size k if and only if there exists a non-contractible edge e in G such that G • e has a (s, t)-CVS of size k − 1.Proof: If: Let S be a (s, t)-CVS of size k in G. Since G[S] is connected, there exists u, v ∈ S such that {u, v} ∈ E(G).Since the cardinality of S is same as the (s, t)-vertex connectivity, the edge e = {u, v} is non-contractible.Moreover, contracting e leaves a graph G • e in which S = (S \ {u, v}) ∪ {z uv } is a vertex separator, where z uv is a new vertex created due to the contraction of {u, v}.Since G[S] is connected and any edge contraction does not disconnect a subgraph which is already connected, G[S ] is a (s, t)-CVS of size k − 1.Therefore, the necessity follows.Only if: Let S be a (s, t)-CVS of size k − 1 in G • e.Clearly, z uv ∈ S, the vertex corresponding to the contraction of the edge {u, v}.In G, S = (S \ {z uv }) ∪ {u, v} is a (s, t)-CVS of size k.Therefore, the sufficiency follows.The above combinatorial observation together with DECIDE-(s,t)-CVS(), we obtain a polynomial-time algorithm to find a minimum (s, t)-CVS of size k and is presented in Algorithm 1.Lemma 5 Let G be a chordality 4 graph having a (s, t)-CVS of size k.Algorithm 2 outputs a k-sized (s, t)-CVS in polynomial time.Proof: The proof of this lemma follows from the fact that Algorithm 2 is an implementation of Lemma 4. The main purpose of Lemma 4 is to ensure that there is no backtracking on an edge e whose contraction reduces the (s, t)-vertex connectivity by 1. Algorithm 1 A Polynomial-time Algorithm to find a minimum (s, t)-CVS in Chordality 4 graphs 1: Input: Chordality 4 graph G with the (s, t)-vertex connectivity k 2: If k = 1 then simply output any cut-vertex in G 3: if DECIDE-(s,t)-CVS(G,k) returns 'NO' then 4: