# Cycle-connected mixed graphs and related problems

## Abstract

In this paper, motivated by applications of vertex connectivity of digraphs or graphs, we consider the cycle-connected mixed graph (CCMG, for short) problem, which is in essence different from the connected mixed graph (CMG, for short) problem, and it is modelled as follows. Given a mixed graph $$G=(V,A\cup E)$$, for each pair $$\{u, v\}$$ of two distinct vertices in G, we are asked to determine whether there exists a mixed cycle C in G to contain such two vertices u and v, where C passes through its arc (xy) along the direction only from x to y and its edge xy along one direction either from x to y or from y to x. Particularly, when the CCMG problem is specialized to either digraphs or graphs, we refer to the related version of the CCMG problem as either the cycle-connected digraph (CCD, for short) problem or the cycle-connected graph (CCG, for short) problem, respectively, where such a graph in the CCG problem is called as a cycle-connected graph. Similarly, we consider the weakly cycle-connected (in other words, circuit-connected) mixed graph (WCCMG, for short) problem, only substituting a mixed circuit for a mixed cycle in the CCMG problem. Moreover, given a graph $$G=(V,E)$$, we define the cycle-connectivity $$\kappa _c(G)$$ of G to be 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; Furthermore, for each pair $$\{s, t\}$$ of two distinct vertices in G, we denote by $$\kappa _{sc}(s,t)$$ the maximum number of internally vertex-disjoint cycles in G to only contain such two vertices s and t in common, then we define the strong cycle-connectivity $$\kappa _{sc}(G)$$ of G to be the smallest of these numbers $$\kappa _{sc}(s,t)$$ among all pairs $$\{s, t\}$$ of distinct vertices in G. We obtain the following three results. (1) Using a transformation from the directed 2-linkage problem, which is NP-complete, to the CCD problem, we prove that the CCD problem is NP-complete, implying that the CCMG problem still remains NP-complete, and however, we design a combinatorial algorithm in time $$O(n^2m)$$ to solve the CCG problem, where n is the number of vertices and m is the number of edges of a graph $$G=(V,E)$$; (2) We provide a combinatorial algorithm in time O(m) to solve the WCCMG problem, where m is the number of edges of a mixed graph $$G=(V,A\cup E)$$; (3) Given a graph $$G=(V,E)$$, we present twin combinatorial algorithms to compute cycle-connectivity $$\kappa _c(G)$$ and strong cycle-connectivity $$\kappa _{sc}(G)$$, respectively.

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## Acknowledgements

The author is indeed grateful to the reviewers for their insightful comments and for their suggested changes that improve the presentation greatly.

## Funding

This paper is fully supported by the National Natural Science Foundation of China [Nos.11861075, 12101593] and Fundamental Research Funds for the Central Universities [No.buctrc202219].

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Correspondence to Junran Lichen.

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Lichen, J. Cycle-connected mixed graphs and related problems. J Comb Optim 45, 53 (2023). https://doi.org/10.1007/s10878-022-00979-3

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• DOI: https://doi.org/10.1007/s10878-022-00979-3

### Keywords

• Combinatorial optimization
• Cycle-connected mixed graphs
• Circuit-connected mixed graphs
• Cycle-connectivity
• Combinatorial algorithms