Elsevier

Theoretical Computer Science

Volume 934, 23 October 2022, Pages 47-64
Theoretical Computer Science

On the complexity of singly connected vertex deletion

https://doi.org/10.1016/j.tcs.2022.08.012Get rights and content

Abstract

A digraph D is singly connected if for all ordered pairs of vertices u,vV(D), there is at most one path in D from u to v. In this paper, we study the Singly Connected Vertex Deletion (SCVD) problem: Given an n-vertex digraph D and a positive integer k, does there exist a set SV(D) such that |S|k and DS is singly connected? This problem may be seen as a directed counterpart of the (Undirected) Feedback Vertex Set problem, as an undirected graph is singly connected if and only if it is acyclic. SCVD is known to be NP-hard on general digraphs. We study the complexity of SCVD on various classes of digraphs such as tournaments, and various generalisations of tournaments such as digraphs of bounded independence number, in- and out-tournaments and local tournaments. We show that unlike the Feedback Vertex Set on Tournaments (FVST) problem, SCVD is polynomial-time solvable on tournaments. In addition, we show that SCVD is polynomial-time solvable on digraphs of bounded independence number, and on the class of acyclic local tournaments. We also study the parameterized complexity of SCVD, with k as the parameter, on the class of in-tournaments. And we show that on in-tournaments, SCVD admits a fixed-parameter tractable algorithm and a quadratic vertex kernel. We also show that on the class of local tournaments, which is a sub-class of in-tournaments, SCVD admits a linear vertex kernel.

Introduction

A digraph D is said to be singly connected if for every (ordered) pair of vertices u and v of D, there is at most one (directed) path in D from u to v. In this paper, we study the Singly Connected Vertex Deletion (SCVD, for short) problem, where the goal is to test if a given digraph can be made singly connected by deleting a few vertices. This problem may be seen as a directed counterpart of the Feedback Vertex Set problem. To see this, let us first define undirected singly connected graphs. An undirected graph G is said to be singly connected if for every pair of vertices u and v of G, there is at most one path in G between u and v. But note that an undirected graph is singly connected if and only if it is acyclic. So, the problem of checking whether it is possible to delete at most k vertices from a given graph to make it singly connected is the same as the problem of checking whether it is possible to delete at most k vertices to make a graph acyclic. This precisely is the Feedback Vertex Set (FVS) problem. (A feedback vertex set of a graph is a set of vertices whose deletion will render the graph acyclic.) The complexity of FVS has been studied extensively [3], [9], [11], [15], [16], [17], [22], [26], [31], [32], [36], [37], [38], [39], [42]. FVS, for instance, was one of Karp's 21 NP-hard problems [33]. As for its algorithmic tractability, FVS is fixed-parameter tractable (when parameterized by the solution size) [26], and it admits a quadratic vertex kernel [45]. FVS also admits constant factor approximation algorithms [3], [9], [19], [29].

Coming back to digraphs, the Directed Feedback Vertex Set (DFVS) problem asks if a given digraph can be made acyclic by deleting at most k vertices. Naturally, this problem has been deemed the appropriate directed counterpart of Feedback Vertex Set, and has been studied in the frameworks of approximation algorithms [44] and parameterized algorithms [18]. Although the parameterized complexity of DFVS had been raised as an open problem since the emergence of parameterized algorithms in the early 90s [25], [27], it was settled only in 2008 by Chen et al. [18]. They showed that the problem admits a 4kk!nO(1) time algorithm, and hence is fixed-parameter tractable when parameterized by k. The kernelization complexity of the problem proved to be even more challenging. While the question whether DFVS (parameterized by k) admits a polynomial kernel still remains unresolved, several attempts have been made to study the kernelization complexity of “DFVS-adjacent” problems. These include studying the problem with larger parameters [10], [40], restricting the input digraph to smaller classes [1], [8] and imposing more conditions on the acyclic digraph that results from the deletion of a feedback vertex set [2], [41].

While FVS and DFVS generated a large volume of literature, the SCVD problem, already known to be NP-hard [24], received little attention from the parameterized complexity community. In this paper, as a first step, we start an investigation into the complexity of SCVD on various classes of digraphs such as tournaments, local tournaments, digraphs of bounded independence number etc. We formally define the problem below.

As observed earlier, an undirected graph is singly connected if and only if it is acyclic. But notice that this property does not hold for digraphs. A directed cycle, for instance, is singly connected. And consider a digraph on 3 vertices, say, x,y and z, and with arcs (x,y),(y,z) and (x,z). This digraph, while acyclic, is not singly connected. It is not surprising then that SCVD and DFVS show markedly different behaviour. This is perhaps best illustrated by the fact that while DFVS is NP-hard on tournaments, we show that SCVD is polynomial-time solvable on tournaments (Lemma 2). This difference in behaviour appears even starker when we consider the fact that these two problems require that “obstructions” with a “similar structure” be hit. Notice that obstructions to an acyclic tournament are directed triangles, i.e., all triplets of vertices x,y and z with arcs (x,y),(y,z) and (z,x), whereas obstructions to a singly connected tournament are all triplets of vertices x,y and z with arcs (x,y),(y,z) and (x,z) (see Fig. 1).

A digraph D is not singly connected if and only if there exists a pair of vertices u and v such that D contains two paths from u to v. It is not difficult to see that a digraph D is not singly connected if and only if there exists a pair of vertices u and v such that D contains two internally vertex disjoint paths from u to v (Lemma 1). Two internally vertex disjoint paths between a pair of vertices of a digraph constitute a cycle in the underlying undirected graph. That is, the obstructions to a singly connected digraph are cycles in the underlying undirected graph. But notice that not every cycle in the underlying undirected graph is necessarily an obstruction. Thus both DFVS and SCVD require us to examine if a subset of the cycles in the underlying undirected graph can be hit with a few vertices.

Our contribution. We study the SCVD problem on several well-studied classes of digraphs such as tournaments, α-bounded digraphs, local tournaments, etc.

A digraph D is said to be a tournament if for every pair of vertices u and v of D, exactly one of the arcs (u,v) and (v,u) is present in D. The class of α-bounded digraphs were introduced by Fradkin and Seymour [28] as a generalisation of tournaments. For a fixed positive integer α, a digraph D is said to be α-bounded if the size of a maximum independent set of the underlying undirected graph of D is at most α. Note that tournaments are 1-bounded digraphs. Local tournaments are yet another generalisation of tournaments. A digraph D is said to be an in-tournament (resp. out-tournament) if for every vertex v of D, the set of in-neighbours (resp. out-neighbours) of v induces a tournament. A digraph D is said to be a local tournament if it is both an in-tournament and an out-tournament. A digraph D is said to be an acyclic local tournament if D is both an acyclic digraph and a local tournament. (See Fig. 2. And see, for example, the chapter on locally semi-complete digraphs [5] in the monograph edited by Bang-Jensen and Gutin [7] for a survey of literature on these classes of digraphs.)

We show that Singly Connected Vertex Deletion

  • is polynomial-time solvable on tournaments and α-bounded digraphs,

  • is polynomial-time solvable on acyclic local tournaments,

  • has a 2knO(1) algorithm and an O(k2) vertex kernel on in- and out-tournaments, and

  • has an O(k) vertex kernel on local tournaments.1

The polynomial-time solvability of SCVD on tournaments follows from a simple observation that no tournament with at least four vertices can be singly connected. A similar result holds for α-bounded digraphs as well: No α-bounded digraph with at least 2α2+3α+1 vertices can be singly connected. To prove this result, we use the Gallai-Milgram theorem [30], which says that the vertices of a digraph D can be covered by a collection of pairwise vertex-disjoint paths such that the number of paths does not exceed the size of a maximum independent set of the underlying undirected graph of D. On acyclic local tournaments, we design a polynomial-time algorithm that computes a minimum-sized vertex subset whose deletion will make the digraph singly connected. Our algorithm uses the fact that every connected local tournament has a Hamiltonian path [4], which in turn implies that every connected acyclic local tournament has a unique topological ordering. We show that SCVD on in-tournaments (and out-tournaments) can be reduced to the 3-Hitting Set problem, and thus the problem admits a simple 3knO(1) time branching algorithm and an O(k2) vertex kernel. But we use the technique of iterative compression to design a 2knO(1) algorithm for SCVD on in- and out-tournaments. Finally, our O(k) vertex kernel for SCVD on local tournaments relies on the fact that for a local tournament D and a set of vertices SV(D) such that DS is singly connected, no vertex in S can have more than a constant number of neighbours in V(D)S.

Related work on singly-connected digraphs. As noted above, the SCVD problem was shown to be NP-hard by Dietzfelbinger and Jaberi [24]. The reduction in [24] in fact shows that the problem is NP-hard even on acyclic digraphs. Their work shows that the arc-deletion version of the problem is also NP-hard, i.e., the problem of testing whether a given digraph can be made singly connected by deleting at most a given number of arcs. As for recognising singly connected digraphs, i.e., the problem of testing whether a given digraph is singly connected, Buchsbaum and Carlisle [13] gave an algorithm that runs in O(n2) time, where n is the number of vertices in the input digraph. Khuller [34], [35] gave another O(n2) algorithm for this problem. Dietzfelbinger and Jaberi [24] presented a refined version of the algorithm of Buchsbaum and Carlisle [13] that runs in time O(st+m), where m is the number of arcs, and s and t respectively are the number of sources and sinks in the input digraph.

Section snippets

Preliminaries

For a positive integer n, we denote the set {1,2,,n} by [n]. Let S be a finite set, and let σ be a permutation of the elements of S. For x,yS, we write x<σy to mean that x appears before y in σ. And we write xσy to mean that either x=y or x<σy.

Digraphs. For a digraph D, V(D) denotes the vertex set and A(D) denotes the arc set of D. For a vertex vV(D), ND+(v) denotes the set of all out-neighbours of v, and ND(v) denotes the set of all in-neighbours of v, that is, ND+(v)={uV(D)|(v,u)A(D)}

Singly connected vertex deletion on α-bounded digraphs and acyclic local tournaments

In this section, we study the optimisation version of SCVD restricted to α-bounded digraphs and acyclic local tournaments, and prove that the problem is polynomial time solvable on both these classes of digraphs. That is, we consider the following problem.

Singly connected vertex deletion on in-tournaments

In this section, we design an algorithm for SCVD on in-tournaments that runs in time 2knO(1). We use the technique of iterative compression, introduced by Reed, Smith and Vetta [43] to design this algorithm. We also show that SCVD on in-tournaments admits a kernel with O(k2) vertices.

Recall that a digraph D is said to be an in-tournament if for all vertices vV(D), D[ND(v)] is a tournament. Recall also that the class of in-tournaments is hereditary. We first prove the following preparatory

A linear vertex kernel for SCVD on local tournaments

In this section, we prove that SCVD admits a linear vertex kernel on local tournaments. Specifically, we prove the following theorem.

Theorem 6

SCVD on local tournaments admits a kernel with O(k) vertices.

Let (D,k) be an instance of SCVD, where D is a local tournament. The basis of our kernelization algorithm is Lemma 12, which says that an in-tournament (and hence a local tournament) is singly connected if and only if it does not contain an acyclic triangle as a subgraph. We introduce the following

Conclusion

We studied the SCVD problem on various classes of digraphs such as tournaments, α-bounded digraphs, acyclic local tournaments, in-tournaments and local tournaments. Our algorithm for SCVD on in-tournaments runs in time 2knO(1). It remains to be seen if this runtime is optimal or can be improved. In particular, as noted in Footnote 1, it is open whether SCVD is NP-hard or polynomial time solvable on in-tournaments. Another class of digraphs that one could consider is the class of locally

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank Janosch Fuchs for sharing his tikz template, which we used for drawing the figures in this paper.

Funding

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant no. 819416), and the Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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    An extended abstract of this work (under the same title) [21] has been published in the proceedings of the 31st International Workshop on Combinatorial Algorithms (IWOCA), 2020. This work was done while the first four authors were at The Institute of Mathematical Sciences, HBNI, Chennai, India.

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