On the Computational Complexity of Vertex Integrity and Component Order Connectivity

Abstract

The Weighted Vertex Integrity (wVI) problem takes as input an n-vertex graph G, a weight function w : V (G) → N, and an integer p. The task is to decide if there exists a set X ⊆ V (G) such that the weight of X plus the weight of a heaviest component of G − X is at most p. Among other results, we prove that:(1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight 1;(2) wVI can be solved in O(p p+1 n) time;(3) wVI admits a kernel with at most p 3 vertices.Result (1) 1997), stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an n-vertex graph G, a weight function w : V (G) → N, and two integers k and ℓ, and the task is to decide if there exists a set X ⊆ V (G) such that the weight of X is at most k and the weight of a heaviest component of G − X is at most ℓ. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We obtain several classical and parameterized complexity results on the wCOC problem, uncovering interesting similarities and differences between wCOC and wVI. We prove, among other results, that: (4) wCOC can be solved in O(min{k, ℓ} · n 3 ) time on interval graphs, while the unweighted version can be solved in O(n 2 ) time on this graph class;(5) wCOC is W[1]-hard on split graphs when parameterized by k or by ℓ;(6) wCOC can be solved in 2 O(k log ℓ) n time;(7) wCOC admits a kernel with at most kℓ(k + ℓ) + k vertices.We also show that result (6) is essentially tight by proving that wCOC cannot be solved in 2 o(k log ℓ) n O (1) time, even when restricted to split graphs, unless the Exponential Time Hypothesis fails.

Partitioning a Graph into Small Pieces with Applications to Path Transversal

“…Our reduction is close to that of Drange et al [12] who reduced Clique to k-Vertex Separator to prove W[1]-hardness. Formally, we introduce another problem called Minimum k-Edge Coverage.…”

“…Our result on k-Edge Separator improves the best previous graph partitioning algorithm [24] for small k. Our result on k-Vertex Separator improves the simple (k + 1)-approximation from HVC [3]. When OPT > k, the running time 2 O(k) n O(1) is faster than the lower bound k Ω(OPT) n Ω(1) for exact algorithms assuming the Exponential Time Hypothesis [12]. While the running time of 2 O(k) n O(1) for k-Vertex Separator seems unsatisfactory, we show that the superpolynomial dependence on k may be needed to achieve a polylogarithmic approximation ratio, based on hardness of Densest k-Subgraph.…”

“…When OPT k, it runs even faster than the time lower bound k Ω(OPT) n Ω(1) for the exact algorithm assuming the Exponential Time Hypothesis [12].…”

“…For small values of k and OPT, the complexity of exact algorithms has been studied in terms of their fixed parameter tractability (FPT). While the trivial algorithm takes n O(OPT) time to find the exact solution for k-Vertex Separator, Drange et al [12] presented an exact algorithm that runs in time k O(OPT) n, so the problem is in FPT when parameterized by both k and OPT. They complemented their result by showing that the problem is W[1]-hard when parameterized by OPT or k. They also showed that any exact algorithm that runs in time…”

Subgraph Isomorphism on Graph Classes that Exclude a Substructure

“…We now check whether Q has vertex integrity at most 3k + 3, which can be done in FPT time parameterized by k [10]. If this is not the case, then Q G. If Q has vertex integrity at most 3k + 3, then we can apply Theorem 4.1.…”