Abstract
Discrete Algorithms A graph containment problem is to decide whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed. We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we consider the problems that are to test whether a given host graph contains a fixed target graph as a contraction.
Highlights
Whether or not a graph G contains a graph H depends on the notion of containment we use; in the literature several natural definitions have been studied
For any fixed H, the problems H-DISSOLUTION, H-SUBGRAPH ISOMORPHISM, H-INDUCED SUBGRAPH ISOMORPHISM, H-SPANNING SUBGRAPH ISOMORPHISM, and H-GRAPH ISOMORPHISM can be solved in polynomial time by brute force
A celebrated result by Robertson and Seymour [22] states that the problems H-MINOR and H-TOPOLOGICAL MINOR can be solved in cubic time and polynomial time, respectively, for every fixed graph H
Summary
A graph containment problem is that of deciding whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we show that the H-CONTRACTIBILITY problem, which asks whether a given host graph contains a fixed target graph H as a contraction, is polynomial-time solvable on claw-free graphs when H is the 4-vertex path P4, whereas on general graphs P4-CONTRACTIBILITY is known to be NP-complete
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