A Characterization of Connected (1,2)-Domination Graphs of Tournaments

Recently. Hedetniemi et aI. introduced (1,2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1,2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1,2)-dominating graph of D, dom1,2 (D), is defined to be the graph G = (V, E ) , where V (G) = V (D), and xy is an edge of G whenever x and y form a (1,2)-dominating pair in D. In this paper, we characterize all connected graphs that can be (I, 2)-dominating graphs of tournaments

Digraphs with Isomorphic Underlying and Domination Graphs: Pairs of Paths

A domination graph of a digraph D, dom (D), is created using thc vertex set of D and edge uv ϵ E (dom (D)) whenever (u, z) ϵ A (D) or (v, z) ϵ A (D) for any other vertex z ϵ A (D). Here, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. Specifically, digraphs are completely characterized where UGc (D) is the union of two disjoint paths

Local Out-Tournaments with Upset Tournament Strong Components I: Full and Equal {0,1}-Matrix Ranks

A digraph D is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corresponding results for biclique covers and partitions of the digraph are provided

Characterization of digraphs with equal domination graphs and underlying graphs

A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u,v}∈E[dom(D)] whenever (u,z)∈A(D) or (v,z)∈A(D) for every other vertex z∈V(D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)=dom(D). The maximum and minimum number of single arcs in these digraphs, and their characteristics, is given