On complexity of substructure connectivity and restricted connectivity of graphs

TL;DR

This paper studies the computational complexity of four refined graph-connectivity notions: structure connectivity $\kappa(G; H)$, substructure connectivity $\kappa^s(G; H)$, $k$-restricted connectivity $\kappa_k(G)$, and $R^h$-restricted connectivity $\kappa^h(G)$. It proves NP-completeness via reductions: from Dominating Set on planar graphs with maximum degree $3$ to $K_{1,M}$-substructure connectivity (yielding NP-completeness of structure connectivity as a corollary), and from Non-Monotone 2-3Sat to the three-set separator problems that underpin $k$-RC and RhC. The results imply intractability of these generalized connectivity problems in general graphs and motivate parameterized analyses or study on restricted graph families. Together, they clarify the computational limits of network reliability measures and suggest directions for future work on bounds and fixed-parameter algorithms.

Abstract

The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose deletion disconnects $G$ and leaves each component of $G-S$ with more than $k$ vertices (resp. $δ(G-S)\geq h$). In contrast, structure (substructure) connectivity of $G$ is defined as the minimum number of vertex-disjoint subgraphs whose deletion disconnects $G$. As generalizations of the concept of connectivity, structure (substructure) connectivity, restricted connectivity and $R^h$-restricted connectivity have been extensively studied from the combinatorial point of view. Very little is known about the computational complexity of these variants, except for the recently established NP-completeness of $k$-restricted edge-connectivity. In this paper, we prove that the problems of determining structure, substructure, restricted, and $R^h$-restricted connectivity are all NP-complete.

Figures (2)

• Figure 1: The graph $G'$ constructed for proving NP-completeness of the substructure connectivity.
• Figure 2: An RCTSVS instance constructing from a Non-Monotone 2-3Sat instance in which $U=\{u_1,u_2,u_3,u_4\}$, $C=\{\{u_1,\overline{u}_2\},\{\overline{u}_1,u_3,\overline{u}_4\},\{u_2,\overline{u}_3,u_4\}\}$. Here $s=n+2m+p=9$.

Theorems & Definitions (13)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4