Wyatt J. Desormeaux

Total domination critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.

Total domination stable graphs upon edge addition

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs.

Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.

An extremal problem for total domination stable graphs upon edge removal

A connected graph is total domination stable upon edge removal, if the removal of an arbitrary edge does not change the total domination number. We determine the minimum number of edges required for a total domination stable graph in terms of its order and total domination number.

Total Domination in Graphs with Diameter 2

The total domination number γt (G) of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, γt (G) ≤ 1 + √ n ln(n). This bound is optimal in the sense that given any > 0, there exist graphs G with diameter 2 of all sufficiently large even orders n such that γt (G) > ( 1 4+ ) √ n ln(n). C

Neighborhood-restricted [≤ 2] -achromatic colorings

A (closed) neighborhood-restricted ? 2 -coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood, that is, for every vertex v in G , the vertex v and its neighbors are in at most two different color classes. The ? 2 -achromatic number is defined as the maximum number of colors in any ? 2 -coloring of G . We study the ? 2 -achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.

Bounds on the Global Domination Number

Abstract A set S of vertices in a graph G is a global dominating set of G if S simultaneously dominates both G and its complement Ḡ. The minimum cardinality of a global dominating set of G is the global domination number of G. We determine bounds on the global domination number of a graph and relationships between it and other domination related parameters.

Domination Edge Lift Critical Trees

Abstract Let uxv be an induced path with center x in a graph G. The edge lifting of uv off x is defined as the action of removing edges ux and vx from the edge set of G, while adding the edge uv to the edge set of G. We study trees for which every possible edge lift changes the domination number. We show that there are no trees for which every possible edge lift decreases the domination number. Trees for which every possible edge lift increases the domination number are characterized.

Distribution centers in graphs

Abstract For a graph G = ( V , E ) and a set S ⊆ V , the boundary of S is the set of vertices in V ∖ S that have a neighbor in S . A non-empty set S ⊆ V is a distribution center if for every vertex v in the boundary of S , v is adjacent to a vertex in S , say u , where u has at least as many neighbors in S as v has in V ∖ S . The distribution center number of a graph G is the minimum cardinality of a distribution center of G . We introduce distribution centers as graph models for supply–demand type distribution. We determine the distribution center number for selected families of graphs and give bounds on the distribution center number for general graphs. Although not necessarily true for general graphs, we show that for trees the domination number and the maximum degree are upper bounds on the distribution center number.

Connected Fair Domination in Graphs

In this paper, we introduce the notion of connected fair domination in graphs. A connected fair dominating set in a graph G (or \(\mathsf {CFD}\)-set) is a dominating set S such that \(\langle S \rangle \) is connected in G and all vertices not in S are dominated by the same number of vertices from S, i.e., every two vertices not in S has the same number of neighbours in S. The connected fair domination number of G (\(\mathsf {cfd}(G)\)) is the minimum cardinality of a \(\mathsf {CFD}\)-set in G. Apart from finding \(\mathsf {cfd}(G)\) for some standard graphs G, we proved various bounds on \(\mathsf {cfd}(G)\) in terms of order and some other graph parameters of G.