TY - JOUR

T1 - MAXIMAL HEREDITARY KONIG-EGERVARY SET-SYSTEMS

AU - Jarden, A

N1 - Times Cited in Web of Science Core Collection: 2 Total Times Cited: 2 Cited Reference Count: 6

PY - 2019

Y1 - 2019

N2 - A Konig-Egervary graph is a graph G satisfying alpha(G) + mu(G) = vertical bar V(G)vertical bar, where alpha(G) is the cardinality of a maximum independent set and mu(G) is the matching number of G. Such graphs are those that admit a matching between V(G) - boolean AND Gamma and boolean OR Gamma, where Gamma is a set-system comprised of maximum independent sets satisfying vertical bar boolean AND Gamma'vertical bar + vertical bar boolean OR Gamma'vertical bar = 2 alpha(G) for every set-system Gamma' subset of Gamma; we refer to set-systems satisfying this equality hereditarily as hereditary Konig-Egervaty set-systems (HKE set-systems, hereafter). In the current paper, we study the maximal HKE set-systems and invoke characterizations of HKE set-systems. We solve a problem of the author with Levit and Mandrescu, proving that a set-system is HKE if and only if it satisfies the above equality and is included in Omega(G) for some graph G. Generally, we cannot reconstruct the graph from Omega(G), the set of maximum independent sets. But we prove that if alpha(G) is a maximal HKE set-system and V(G) = boolean AND Omega(G), then G is a specific bipartite graph.

AB - A Konig-Egervary graph is a graph G satisfying alpha(G) + mu(G) = vertical bar V(G)vertical bar, where alpha(G) is the cardinality of a maximum independent set and mu(G) is the matching number of G. Such graphs are those that admit a matching between V(G) - boolean AND Gamma and boolean OR Gamma, where Gamma is a set-system comprised of maximum independent sets satisfying vertical bar boolean AND Gamma'vertical bar + vertical bar boolean OR Gamma'vertical bar = 2 alpha(G) for every set-system Gamma' subset of Gamma; we refer to set-systems satisfying this equality hereditarily as hereditary Konig-Egervaty set-systems (HKE set-systems, hereafter). In the current paper, we study the maximal HKE set-systems and invoke characterizations of HKE set-systems. We solve a problem of the author with Levit and Mandrescu, proving that a set-system is HKE if and only if it satisfies the above equality and is included in Omega(G) for some graph G. Generally, we cannot reconstruct the graph from Omega(G), the set of maximum independent sets. But we prove that if alpha(G) is a maximal HKE set-system and V(G) = boolean AND Omega(G), then G is a specific bipartite graph.

KW - Konig-Egervary graphs

KW - independent sets

KW - set-systems

U2 - 10.17654/DM022020195

DO - 10.17654/DM022020195

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SN - 0974-1658

VL - 22

SP - 195

EP - 219

JO - Advances and Applications in Discrete Mathematics

JF - Advances and Applications in Discrete Mathematics

IS - 2

ER -