Paper Info
Title
Total Roman Domination Edge-Supercritical And Edge-Removal-Supercritical Graphs
Abstract
A total Roman dominating function on a graph G is a function f : V (G) -> {0, 1, 2} such that every vertex v with f(v) = 0 is adjacent to i some vertex u with f(u) = 2, and the subgraph of G induced by the set of all vertices w such that f(w) > 0 has no isolated vertices. The weight of f is Sigma(v is an element of V) (G) f(v). The total Roman domination number gamma(tR)(G) is the minimum weight of a total Roman dominating function on G. A graph G is k-gamma(tR)-edge-critical if gamma(tR)(G + e) <.tR(G) = k for every edge e is an element of E(<(G)over bar>) not equal phi, and k- gamma(tR)-edge-supercritical if it is k-gamma(tR)-edge-critical and gamma(tR)(G + e) = gamma(tR)(G) - 2 for every edge e is an element of E((G) over bar) not equal phi. A graph G is k-gamma(tR)-edge-stable if gamma(tR)(G + e) = gamma(tR)(G) = k for every edge e is an element of E((G) over bar) or E((G) over bar) = phi. For an edge e is an element of E(G) incident with a degree 1 vertex, we define gamma(tR)(G - e) = infinity. A graph G is k- gamma(tR)-edge-removal-critical if gamma(tR)(G - e) > gamma(tR)(G) = k for every edge e is an element of E(G), and k-gamma(tR)-edge-removal-supercritical if it is k-gamma(tR)- edge-removal-critical and gamma(tR)(G-e) >= gamma(tR)(G) + 2 for every edge e is an element of E(G). A graph G is k-gamma tR-edge- removal-stable if gamma(tR)(G - e) = gamma(tR)(G) = k for every edge e is an element of E (G). We investigate connected gamma(tR)-edge-supercritical graphs and exhibit infinite classes of such graphs. In addition, we characterize gamma(tR)-edge-removal-critical and gamma(tR)-edge-removal-supercritical graphs. Furthermore, we present a connection between k-gamma tR-edge-removal-supercritical and k-gamma tR-edge-stable graphs, and similarly between k-gamma tR-edge-supercritical and k-gamma tR-edge-removal-stable graphs.
Year
Venue
DocType
2020
AUSTRALASIAN JOURNAL OF COMBINATORICS
Journal
Volume
ISSN
Citations 
78
2202-3518
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Mynhardt C. M.100.34
Ogden S. E. A.200.34