A forbidden subgraph study for cut problems on graphs permitting loops and multiedges

TL;DR

This paper extends the C123 framework to graph problems on multigraphs and partially reflexive graphs, focusing on Multigraph Matching Cut, Multigraph $d$-Cut, and Partially Reflexive Stable Cut. It proves that Multigraph Matching Cut and Multigraph $d$-Cut are C123-problems, while Partially Reflexive Stable Cut is not a C123-problem but can mimic C123-like hardness on certain forbidden-subgraph sets; the authors identify a boundary phenomenon governed by pendant subdivisions of nets and $H_1$. The results include a complete dichotomy for MMC and MD-Cut on ${\\cal H}$-subgraph-free graphs and nuanced tractability for Partially Reflexive Stable Cut depending on the forbidden-subgraph family, including a key boundary case around $H_1^{2,2,2,1}$ exhibiting polynomial-time solvability with $C_3$-free graphs. The work demonstrates how enriching inputs with loops and multiedges interacts with forbidden-subgraph classifications, offering new insights and open questions on the precise tractability frontier.

Abstract

We take the recently-introduced C123-framework, for the study of (simple) graph problems restricted to inputs specified by the omission of some finite set of subgraphs, to more general graph problems possibly involving self-loops and multiedges. We study specifically the problems Partially Reflexive Stable Cut and Multigraph Matching Cut in this connection. When one forbids a single (simple) subgraph, these 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 are C123-problems, already Partially Reflexive Stable Cut fails to be. 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.

Figures (6)

• Figure 1: The clause $C_1=(\bar{x_1} \lor x_2 \lor x_3$) in the case $k=1$.
• Figure 2: The clause $C_1=(\bar{x_1} \lor x_2 \lor x_3)$
• Figure 3: $H_1^{2,2,2,1}$
• Figure 4: Case 1, $x=a',x'=c'$. Left: If the edge $(d,c')$ exists, at least one of the dotted edges must exist. Right: Given $(d,c')$ is not an edge, both the edges $(a,d)$ and $(a',q')$ must exist.
• Figure 5: Case 2, $x=a',x'=p'$. The edges $(a,c)$ and $(a',d)$ must exist else this is case 1.

Theorems & Definitions (36)

• theorem 1: JMOPPSV
• theorem 2
• theorem 3
• theorem 4
• proposition 1
• proof
• proposition 2: MatchingCut
• lemma 1
• proof
• proposition 3