Abstract
A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of any one vertex leaves a graph, which is well-covered as well (Staples, 1975). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given independent set. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs.
| Original language | English |
|---|---|
| Pages (from-to) | 261-272 |
| Number of pages | 12 |
| Journal | European Journal of Combinatorics |
| Volume | 80 |
| DOIs | |
| State | Published - Aug 2019 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2018 Elsevier Ltd
Funding
We express our gratitude to Zakir Deniz, who has pointed out that the well-coverednesscondition in Theorem 3.9 (iii) is necessary. Our special thanks are to Tran Nam Trung for: an example that has helped us to formulate Problem 4.4 more precisely; introducing us to Proposition 4.7; and some valuable suggestions that have brought us to the proof of Theorem 3.9 (iv), (v). We also thank the anonymous reviewers for many beneficial comments and remarks, which substantially improved the presentation of the paper. One of them suggested a generalization of Corollary 2.6, which has been incorporated into the text as Proposition 2.5.