In a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $Gamma$ is said to be   $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power or...

Lexicographic product G◦H of two graphs G and H has vertex set V (G)×V (H) and two vertices (u1, v1) and (u2, v2) are adjacent whenever u1u2 ∈ E(G), or u1 = u2 and v1v2 ∈ E(H). If every matching of G of size k can be extended to a perfect matching in G, then G is called k-extendable. In this paper, we study matching extendability in lexicographic product of graphs. The main result is that the l...

We show that obtaining the lexicographically first four coloring of a planar graph is NP–hard. This shows that planar graph four-coloring is not self-reducible, assuming P 6= NP . One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph k-coloring, for k ≥ 4. Resear...

Let x be a vertex of a simple graph G. The vertex-type of x is the lexicographically ordered degree sequence of its neighbors. We call the graph G vertex-oblique if there are no two vertices in V (G) which are of the same vertex-type. We will show that the set of vertex-oblique graphs of arbitrary connectivity is infinite. © 2006 Elsevier B.V. All rights reserved.

Background: In recent years, there has been an increase in the amount and variety of data generated in the field of healthcare, (e.g., data related to the prevalence of contagious diseases in the society). Various patterns of individuals’ relationships in the society make the analysis of the network a complex, highly important process in detecting and preventing the incidence of diseases....