Computing the largest bond of a graph

TL;DR

This paper investigates the computational complexity of finding the largest bond and largest st-bond in graphs, revealing NP-hardness, inapproximability, and fixed-parameter tractability results, thus filling a gap in graph cut research.

Contribution

It establishes the NP-hardness and inapproximability of Largest Bond problems, and provides fixed-parameter algorithms and complexity bounds based on clique-width and solution size.

Findings

Largest Bond is NP-hard even for planar bipartite graphs.

No constant-factor approximation exists unless P=NP.

Problems are fixed-parameter tractable by solution size but lack polynomial kernels.

Abstract

A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of $G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Given $s,t\in V(G)$, an $st$-bond of $G$ is a bond whose removal disconnects $s$ and $t$. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest $st$-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that {\sc Largest Bond} remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless $P = NP$. We also show that {\sc Largest Bond} and {\sc Largest $st$-Bond} on graphs of clique-width $w$ cannot be solved in time $f(w)\times n^{o(w)}$ unless the Exponential Time Hypothesis fails, but they can be solved in time $f(w)\times n^{O(w)}$. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP $\subseteq$ coNP/poly.