P. Kristiansen, S.M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number γa(G), strong defensive alliance number aˆ(G), and global defensive alliance number γa(G). In this paper, we consider relationships between these parameters and the domination number γ(G). For any positive integers a, b, and c satisfying a ≤ c and b ≤ c, there is a graph G with a = a(G), b = γ(G), and c = γa(G). For any positive integers a, b, and c, provided a ≤ b ≤ c and c is not too much larger than a and b, there is a graph G with γ(G) = a, γa(G) = b, and γaˆ(G) = c. Given two connected graphs H1 and H2, where order(H1) ≤ order(H2), there exists a graph G with a unique minimum defensive alliance isomorphic to H1 and a unique minimum strong defensive alliance isomorphic to H2.
The eccentricity of a vertex v of a connected graph G is the distance from v to a vertex farthest from v in G. The center of G is the subgraph of G induced by the vertices having minimum eccentricity. For a vertex v in a 2-edge-connected graph G, the edge-deleted eccentricity of v is defined to be the maximum eccentricity of v in G − e over all edges e of G. The edge-deleted center of G is the subgraph induced by those vertices of G having minimum edge-deleted eccentricity. The edge-deleted central appendage number of a graph G is the minimum difference |V (H)| − |V (G)| over all graphs H where the edgedeleted center of H is isomorphic to G. In this paper, we determine the edge-deleted central appendage number of all trees.
Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the ''AIM Minimum Rank–Special Graphs Work Group'', whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that Z(G) ≤ 2Z(L(G)) for a simple and connected graph G. Further, we show that Z(G) ≤ Z(L(G)) when G is a tree or when G contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems.