Characterizing all nonbipartite well-edge-dominated graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916070717128704 |
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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 of $G$ if every edge in $G$ is either in $F$ or adjacent to an edge in $F$. A graph $G$ is said to be well-edge-dominated if every minimal edge dominating set has the same cardinality. This definition is the edge version of domination in that a set $D\subseteq V(G)$ is a dominating set if every vertex in $G$ is in $D$ or adjacent to a vertex in $D$ and the domination number $γ(G)$ is the minimum cardinality among all dominating sets. In this paper, we complete the characterization of all nonbipartite, well-edge-dominated graphs. In addition, we produce an infinite class of graphs that satisfy the well-known Vizing's conjecture in domination theory that states $γ(G\Box H) \ge γ(G)γ(H)$ where $G\Box H$ is the Cartesian product of $G$ and $H$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_00802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterizing all nonbipartite well-edge-dominated graphs Anderson, Sarah E. Kuenzel, Kirsti Combinatorics 05C69, 05C76, 05C75 Given a graph $G$, a set $F$ of edges is an edge dominating set of $G$ if every edge in $G$ is either in $F$ or adjacent to an edge in $F$. A graph $G$ is said to be well-edge-dominated if every minimal edge dominating set has the same cardinality. This definition is the edge version of domination in that a set $D\subseteq V(G)$ is a dominating set if every vertex in $G$ is in $D$ or adjacent to a vertex in $D$ and the domination number $γ(G)$ is the minimum cardinality among all dominating sets. In this paper, we complete the characterization of all nonbipartite, well-edge-dominated graphs. In addition, we produce an infinite class of graphs that satisfy the well-known Vizing's conjecture in domination theory that states $γ(G\Box H) \ge γ(G)γ(H)$ where $G\Box H$ is the Cartesian product of $G$ and $H$. |
| title | Characterizing all nonbipartite well-edge-dominated graphs |
| topic | Combinatorics 05C69, 05C76, 05C75 |
| url | https://arxiv.org/abs/2606.00802 |