Characterizing all $K_4$-free well-edge-dominated graphs of girth 3
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914238382997504 |
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| author | Anderson, Sarah E. Kuenzel, Kirsti |
| author_facet | Anderson, Sarah E. Kuenzel, Kirsti |
| contents | Given a graph $G$, a set $F$ of edges is an edge dominating set if all edges in $G$ are either in $F$ or adjacent to an edge in $F$. $G$ is said to be well-edge-dominated if every minimal edge dominating set is also minimum. In 2022, it was proven that there are precisely three nonbipartite, well-edge-dominated graphs with girth at least four. Then in 2025, a characterization of all well-edge-dominated graphs containing exactly one triangle was found. In this paper, we characterize all well-edge-dominated graphs that contain a triangle and yet are $K_4$-free. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing all $K_4$-free well-edge-dominated graphs of girth 3 Anderson, Sarah E. Kuenzel, Kirsti Combinatorics 05C69, 05C76, 05C75 Given a graph $G$, a set $F$ of edges is an edge dominating set if all edges in $G$ are either in $F$ or adjacent to an edge in $F$. $G$ is said to be well-edge-dominated if every minimal edge dominating set is also minimum. In 2022, it was proven that there are precisely three nonbipartite, well-edge-dominated graphs with girth at least four. Then in 2025, a characterization of all well-edge-dominated graphs containing exactly one triangle was found. In this paper, we characterize all well-edge-dominated graphs that contain a triangle and yet are $K_4$-free. |
| title | Characterizing all $K_4$-free well-edge-dominated graphs of girth 3 |
| topic | Combinatorics 05C69, 05C76, 05C75 |
| url | https://arxiv.org/abs/2511.06095 |