Complexity Framework For Forbidden Subgraphs V: Beyond Simple Graphs

TL;DR

This paper extends the complexity analysis of graph problems within the C123-framework to more general graphs with multiedges and loops, revealing nuanced differences in problem behaviors based on forbidden subgraphs.

Contribution

It broadens the C123-framework to include multigraphs and self-loops, providing new complexity classifications and highlighting differences in problem behaviors with various forbidden subgraphs.

Findings

Multigraph Matching Cut and Multigraph d-Cut follow known complexity patterns.

Partially Reflexive Stable Cut exhibits more complex behavior with certain forbidden subgraphs.

Some surjective homomorphism problems are polynomial-time solvable on bounded degree graphs.

Abstract

We continue the study of the recently-introduced C123-framework, for (simple) graph problems restricted to inputs specified by the forbidding of some finite set of subgraphs, to more general graph problems possibly involving multiedges and self-loops. We study specifically the problems Multigraph Matching Cut, Multigraph d-Cut and Partially Reflexive Stable Cut in this connection. The last may be seen as a Surjective Homomorphism problem to a path P_3 in which both leaves are looped while the interior vertex is loopless. We consider also another family of Surjective Homomorphism problems to a cycle in which only one vertex is loopless. When one forbids a single (simple) subgraph, our first three problems exhibit the same complexity behaviour as C123-problems, but on finite sets of forbidden subgraphs, the classification appears more complex. While Multigraph Matching Cut and Multigraph d-Cut have the same classification as C123-problems, already Partially Reflexive Stable Cut fails to have. This is witnessed by forbidding as subgraphs both C_3 and H_1. Indeed, the difference of behaviour occurs only around pendant subdivisions of nets and pendant subdivisions of H_1. We examine this area in close detail. Our other Surjective Homomorphism problem, ostensibly somewhat similar to Partially Reflexive Stable Cut, behaves very differently when the input is restricted to some class that is H-subgraph-free. For example, it is solvable in polynomial time on any class of bounded degree. Also, its hardness will never be preserved under any form of edge subdivision.