On 1-Konig-Egervary Graphs
Abstract
Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. Let $\xi(G)$ denote the size of the intersection of all maximum independent sets. It is known that if $\alpha(G)+\mu(G)=n(G)=\left\vert V\right\vert $, then $G$ is a König-Egerváry graph. If $\alpha(G)+\mu(G)=n(G) -1$, then $G$ is a $1$-König-Egerváry graph. If $G$ is not a König-Egerváry graph, and there exists a vertex $v\in V$ (an edge $e\in E$) such that $G-v$ ($G-e$) is König-Egerváry, then $G$ is called a vertex (an edge) almost König-Egerváry graph (respectively). The critical difference $d(G)$ is $\max\{d(I):I\in\mathrm{Ind}(G)\}$, where $\mathrm{Ind}(G)$ denotes the family of all independent sets of $G$. If $A\in\mathrm{Ind}(G)$ with $d\left( X\right) =d(G)$, then $A$ is a critical independent set. Let $diadem (G)=\bigcup\{S:S$ is a critical independent set in $G\}$, and $\varrho_{v}\left( G\right) $ denote the number of vertices $v\in V\left( G\right) $, such that $G-v$ is a König-Egerváry graph. In this paper, we characterize all types of almost König-Egerváry graphs and present interrelationships between them. We also show that if $G$ is a $1$-König-Egerváry graph, then $\varrho_{v}\left( G\right) \leq n\left( G\right) +d\left( G\right) -\xi\left( G\right) -\beta(G)$, where $\beta(G)=\left\vert diadem(G)\right\vert $. As an application, we characterize the $1$-König-Egerváry graphs that become König-Egerváry after deleting any vertex.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2023
- DOI:
- arXiv:
- arXiv:2308.03503
- Bibcode:
- 2023arXiv230803503L
- Keywords:
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- Mathematics - Combinatorics;
- 05C69 (Primary) 05C70 (Secondary);
- G.2.2
- E-Print:
- 17 pages, 10 figures