## 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 (*x*, *y*) 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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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