Graphs with convex domination number close to their order

Abstract

For a connected graph G = (V, E), a set D ⊆ V (G) is a dominating set of G if every vertex in V (G) − D has at least one neighbour in D. The distance d G (u, v) between two vertices u and v is the length of a shortest (u − v) path in G. An (u − v) path of length d G (u, v) is called an (u − v)-geodesic. A set X ⊆ V (G) is convex in G if vertices from all (a − b)-geodesics belong to X for any two vertices a, b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γ con (G) of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.

Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs

“…In a comprehensive study regarding graphs with convex domination number close to their order [8], a dozen of very useful results have been presented. Among all of them, we emphasize a remark and two theorems, which are of interest for our paper.…”

“…[8]). If G = (V, E) is a connected graph with δ(G) ≥ 2 and g(G) ≥ 6, then γ con (G) = |V |, where δ(G) is the minimum vertex degree and g(G) is the length of a shortest cycle in G.…”

“…[8]). For any connected graphs G and H, the inequalityγ con (G)γ con (H) ≤ γ con (G H)holds, where G H denotes the Cartesian product of the graphs G and H.…”

Geodesic Convexity in Graphs

“…Further results involving geodominating invariants related to geodesic convexity in graphs can be found in [2,7,8,13,49,52,53,58,60,77,79,97,109,111,125,126,137,146,147,166,171,173,176,177,181]. …”

Convex universal fixers

“…The convex domination number was defined by Jerzy Topp from the Gdańsk University of Technology in a verbal communication with the first author. In [5], the first results concerning this topic were published and developed in [6] and [7].…”