Gamma-Set Domination Graphs. I: Complete Biorientations of \u3cem\u3eq-\u3c/em\u3eExtended Stars and Wounded Spider Graphs

The domination number of a graph G, γ(G), and the domination graph of a digraph D, dom(D) are integrated in this paper. The γ-set domination graph of the complete biorientation of a graph G, domγ(G) is created. All γ-sets of specific trees T are found, and dom-γ(T) is characterized for those classes

The (1,2)-Step Competition Graph of a Tournament

The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the m-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)-step competition graph of a digraph D, denoted C1,2(D), as the graph on V(D) where {x,y}∈E(C1,2(D)) if and only if there exists a vertex z≠x,y, such that either dD−y(x,z)=1 and dD−x(y,z)≤2 or dD−x(y,z)=1 and dD−y(x,z)≤2. In this paper, we characterize the (1,2)-step competition graphs of tournaments and extend our results to the (i,k)-step competition graph of a tournament

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

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

Kings and Heirs: A Characterization of the (2,2)-domination Graphs of Tournaments

In 1980, Maurer coined the phrase king when describing any vertex of a tournament that could reach every other vertex in two or fewer steps. A (2,2)-domination graph of a digraph D, dom2,2(D), has vertex set V(D), the vertices of D, and edge uv whenever u and v each reach all other vertices of D in two or fewer steps. In this special case of the (i,j)-domination graph, we see that Maurer’s theorem plays an important role in establishing which vertices form the kings that create some of the edges in dom2,2(D). But of even more interest is that we are able to use the theorem to determine which other vertices, when paired with a king, form an edge in dom2,2(D). These vertices are referred to as heirs. Using kings and heirs, we are able to completely characterize the (2,2)-domination graphs of tournaments

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

Observation of Asymmetric Transport in Structures with Active Nonlinearities

A mechanism for asymmetric transport based on the interplay between the fundamental symmetries of parity (P) and time (T) with nonlinearity is presented. We experimentally demonstrate and theoretically analyze the phenomenon using a pair of coupled van der Pol oscillators, as a reference system, one with anharmonic gain and the other with complementary anharmonic loss; connected to two transmission lines. An increase of the gain/loss strength or the number of PT-symmetric nonlinear dimers in a chain, can increase both the asymmetry and transmittance intensities.Comment: 5 pages, 5 figure