Minimal toughness in special graph classes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2018
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| _version_ | 1866917587895451648 |
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| author | Katona, Gyula Y. Varga, Kitti |
| author_facet | Katona, Gyula Y. Varga, Kitti |
| contents | Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1802_00055 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Minimal toughness in special graph classes Katona, Gyula Y. Varga, Kitti Combinatorics 05C99 Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs. |
| title | Minimal toughness in special graph classes |
| topic | Combinatorics 05C99 |
| url | https://arxiv.org/abs/1802.00055 |