On complexity of substructure connectivity and restricted connectivity of graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866915728462970880 |
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| author | Lü, Huazhong Wu, Tingzeng |
| author_facet | Lü, Huazhong Wu, Tingzeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05917 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On complexity of substructure connectivity and restricted connectivity of graphs Lü, Huazhong Wu, Tingzeng Computational Complexity Combinatorics 03D15, 05C40, 68R10 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. |
| title | On complexity of substructure connectivity and restricted connectivity of graphs |
| topic | Computational Complexity Combinatorics 03D15, 05C40, 68R10 |
| url | https://arxiv.org/abs/2110.05917 |