Cycle-Connected Mixed Graphs and Related Problems

Abstract

In this paper, motivated by vertex connectivity of digraphs or graphs, we address the cycle-connected mixed graph (CCMG) problem. Specifically, given a mixed graph $G=(V,A\cup E)$, for every pair x, y of distinct vertices in G, we are asked to find a mixed cycle C in G to contain such two vertices x and y, where C traverses its arc (u, v) along the direction from u to v and its edge uv along one direction either from u to v or from v to u. Similarly, we consider the circuit-connected (or weakly cycle-connected) mixed graph (WCCMG) problem, substituting a mixed circuit for a mixed cycle in the CCMG problem. Whenever the CCMG problem is specialized to either digraphs or graphs, we refer to this version as either the cycle-connected digraph (CCD) problem or the cycle-connected graph (CCG) problem, and we refer to this mixed graph as either a cycle-connected digraph or a cycle-connected graph in related versions. Furthermore, given a graph $G=(V,E)$, the cycle-connectivity, denoted by $\kappa _c(G)$, of G is the smallest number of vertices in G whose deletion causes the reduced subgraph either not to be a cycle-connected graph or to become an isolated vertex; In addition, for every pair x, y of distinct vertices in G, we denote by $\kappa _{sc}(x,y)$ the maximum number of internally vertex-disjoint cycles in G containing x and y, then the strong cycle-connectivity, denoted by $\kappa _{sc}(G)$, of G is the smallest of these numbers $\kappa _{sc}(x,y)$ among all pairs $\{x, y\}$ of distinct vertices in G, i.e., $\kappa _{sc}(G)=\min \{\kappa _{sc}(x,y)~|~x,y\in V\}$.

We obtain the three main results. (1) Using the directed 2-linkage problem which is NP-complete, we prove that the CCD problem is NP-complete, implying that the CCMG problem still remains NP-complete, and however, we can design an exact algorithm in polynomial time to solve the CCG problem; (2) We present a simply exact algorithm in polynomial time to solve the WCCMG problem; (3) Given a graph $G=(V,E)$, we provide twin exact algorithms in polynomial time to compute cycle-connectivity $\kappa _c(G)$ and strong cycle-connectivity $\kappa _{sc}(G)$, respectively.

This paper is supported by the National Natural Science Foundation of China [Nos. 11861075, 12101593].