Vol. 12, No. 8, 2019

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Total Roman domination edge-critical graphs

Chloe Lampman, Kieka (C. M.) Mynhardt and Shannon Ogden

Vol. 12 (2019), No. 8, 1423–1439

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 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 vV (G)f(v). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function on G. A graph G is k-γtR-edge-critical if γtR(G + e) < γtR(G) = k for every edge e E( G ¯), and k-γtR-edge-supercritical if it is k-γtR-edge-critical and γtR(G + e) = γtR(G) 2 for every edge e E( G ¯). We present some basic results on γtR-edge-critical graphs and characterize certain classes of γtR-edge-critical graphs. In addition, we show that, when k is small, there is a connection between k-γtR-edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.

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Roman domination, total Roman domination, total Roman domination edge-critical graphs
Mathematical Subject Classification 2010
Primary: 05C69
Received: 23 July 2019
Accepted: 26 September 2019
Published: 25 October 2019

Communicated by Anant Godbole
Chloe Lampman
Department of Mathematics and Statistics
University of Victoria
Victoria, BC
Kieka (C. M.) Mynhardt
Department of Mathematics and Statistics
University of Victoria
Victoria, BC
Shannon Ogden
Department of Mathematics and Statistics
University of Victoria
Victoria, BC