439 research outputs found
Contraction blockers for graphs with forbidden induced paths.
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pâ„“-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
Reducing the clique and chromatic number via edge contractions and vertex deletions.
We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H
Reducing the clique and chromatic number via edge contractions and vertex deletions
We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H
Contraction blockers for graphs with forbidden induced paths
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pâ„“-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions
We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H
On Blockers and Transversals of Maximum Independent Sets in Co-Comparability Graphs
In this paper, we consider the following two problems: (i) Deletion
Blocker() where we are given an undirected graph and two
integers and ask whether there exists a subset of vertices
with such that , that
is the independence number of decreases by at least after having
removed the vertices from ; (ii) Transversal() where we are given an
undirected graph and two integers and ask whether there
exists a subset of vertices with such that for every
maximum independent set we have . We show that both
problems are polynomial-time solvable in the class of co-comparability graphs
by reducing them to the well-known Vertex Cut problem. Our results generalize a
result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer
Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al.,
Assistance and interdiction problems on interval graphs, Discrete Applied
Mathematics 340, pp. 153-170, 2023]
Blocking Dominating Sets for H-Free Graphs via Edge Contractions
In this paper, we consider the following problem: given a connected graph G, can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP-hard when restricted to {P6, P4 + P2}-free graphs and that it is coNP-hard when restricted to subcubic claw-free graphs and 2P3-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H-free graphs when H is connected
Reducing Graph Transversals via Edge Contractions
For a graph parameter ?, the Contraction(?) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which ? has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where ? is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ? according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ?, which in particular imply that Contraction(?) is co-NP-hard even for fixed k = d = 1 when ? is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when ? is the size of a minimum vertex cover, the problem is in XP parameterized by d
Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions
Let d and k be two given integers, and let G be a graph. Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings: – we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP- hard even if d = k = 1; – in addition we prove that the vertex deletion variant is co-NP- hard for triangle-free graphs even if d = k = 1; – we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees. By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs
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