SECURE POINT SET DOMINATION IN GRAPHS

• Published in: TWMS journal of applied and engineering mathematics Vol. 14; no. 2; p. 605
• Main Authors: Gupta, Purnima, Goyal, Alka
• Format: Journal Article
• Language: English
• Published: Turkic World Mathematical Society 01.04.2024
• Subjects: Fuzzy sets; Graph theory; Mathematical research; Set theory; India
• ISSN: 2146-1147

In this paper, we introduce the notion of secure point-set domination in graphs. A point-set dominating D of graph G is called a secure point-set dominating set if for every vertex u [member of] V - D, there exists a vertex v [member of] D [intersection] N (u) such that (D - {v}) [union]{u} is also a point-set dominating set of G. The minimum cardinality of a secure point-set dominating set is called secure point-set domination number of graph G and will be denoted by [[gamma].sub.spsd](G) (or simply [[gamma].sub.spsd]). For any graph G of order n, [[gamma].sub.spsd](G) [greater than or equal to] 1 and equality holds if and only if G [congruent to] [K.sub.n]. Also, for any graph G of order n, [[gamma].sub.spsd](G) [less than or equal to] n - 1 and equality holds if and only if G [congruent to] [K.sub.1,n-1]. Here we characterize graphs G with [[gamma].sub.spsd](G) = 2. We also establish a family F of 11 graphs such that being F-free is necessary as well as sufficient for a graph G to satisfy [[gamma].sub.spsd](G) = n - 2. Keywords: Domination, Point-Set Domination, Secure Domination, Secure Point-Set Domination, Secure Point-Set Domination Number. AMS Subject Classification: 05C69.