Restrained and Total Restrained Domination of Ladder Graphs - Hemalatha, N. C.; Chandrakala, S. B.; Sooryanarayana, B.; Kumar, M. Vishu (Communications in Mathematics and Applications, 2023)
Telle and Proskurowksi introduced restrained domination as a vertex partition problem in partial \(k\)-tress (Algorithms for vertex partitioning problems on partial \(k\)-trees, SIAM Journal on Discrete Mathematics 10(4) (1997), 529 - 550). For a graph \(G(V,E)\), a restrained domination number is the minimum cardinality of a subset \(\mathfrak{D}\) of \(V\) such that for every vertex \(v\in \bar{\mathfrak{D}}\) there is a vertex in \(\mathfrak{D}\) as well as in \(\bar{\mathfrak{D}}\) adjacent to \(v\). If \(\mathfrak{D}\) satisfies an additional condition that every vertex of \(V\) has a neighbor in \(\mathfrak{D}\), then \(\mathfrak{D}\) is said to be a total restrained dominating set. Minimum cardinality of \(\mathfrak{D}\) is said to be total restrained domination number of graph \(G\). In this paper we have obtained domination, restrained, total and total restrained domination number of some ladder graphs.
Keywords: Domination; Total domination; Restrained domination