| Abstract |
Let G be a graph. A set S 鈯� V (G) is a hop dominating set of G if for every v 鈭� V (G)S, there exists u 鈭� S such that dG(u, v) = 2. The minimum cardinality 纬h(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of G of cardinality 纬h(G) is a 纬h-set of G. A hop dominating set S of G which intersects every 纬h-set of G is a transversal hop dominating set. The minimum cardinality 纬bh(G) of a transversal hop dominating set in G is the transversal hop domination number of G. In this paper, we initiate the study of transversal hop domination. First, we characterize graphs G whose values for 纬bh(G) are either n or n 鈭� 1, and we determine the specific values of 纬bh(G) for some specific graphs. Next, we show that for every positive integers a and b with a 鈮� 2 and b 鈮� 3a, there exists a connected graph G on b vertices such that 纬bh(G) = a. We also show that for every positive integers a and b with 2 鈮� a 鈮� b, there exists a connected graph G for which 纬h(G) = a and 纬bh(G) = b. Finally, we investigate the transversal hop dominating sets in the join and corona of two graphs, and determine their corresponding transversal hop domination numbers. |
Mindanao State
University - Iligan Institute of Technology