{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:50:48Z","timestamp":1759063848228,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a simple graph with vertex set $V(G)$. A set $S\\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \\max \\{d(X): X\\subseteq V(G) \\text{ is an independent set}\\}$ (possibly $A=\\emptyset$). Let $\\text{nucleus}(G)$ and $\\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of Konig-Egervary\u00a0graphs involving $\\text{nucleus}(G)$ and $\\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.<\/jats:p>","DOI":"10.37236\/5580","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:59:28Z","timestamp":1578689968000},"source":"Crossref","is-referenced-by-count":8,"title":["On Some Conjectures Concerning Critical Independent Sets of a Graph"],"prefix":"10.37236","volume":"23","author":[{"given":"Taylor","family":"Short","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,6,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:27:49Z","timestamp":1579238869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i2p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,10]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,3,31]]}},"URL":"https:\/\/doi.org\/10.37236\/5580","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,6,10]]},"article-number":"P2.43"}}