Teresa W. Haynes

Double Roman domination

For a graph G = ( V , E ) , a double Roman dominating function is a function f : V ź { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then vertex v must have at least one neighbor with f ( w ) ź 2 . The weight of a double Roman dominating function f is the sum f ( V ) = ź v ź V f ( v ) , and the minimum weight of a double Roman dominating function on G is the double Roman domination number of G . We initiate the study of double Roman domination and show its relationship to both domination and Roman domination. Finally, we present an upper bound on the double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound.

Bounds on weak roman and 2-rainbow domination numbers

Abstract We mainly study two related dominating functions, namely, the weak Roman and 2-rainbow dominating functions. We show that for all graphs, the weak Roman domination number is bounded above by the 2-rainbow domination number. We present bounds on the weak Roman domination number and the secure domination number in terms of the total domination number for specific families of graphs, and we show that the 2-rainbow domination number is bounded below by the total domination number for trees and for a subfamily of cactus graphs.

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

Roman { 2 } -domination

In this paper, we initiate the study of a variant of Roman dominating functions. For a graph G = ( V , E ) , a Roman { 2 } -dominating function f : V ? { 0 , 1 , 2 } has the property that for every vertex v ? V with f ( v ) = 0 , either v is adjacent to a vertex assigned 2 under f , or v is adjacent to least two vertices assigned 1 under f . The weight of a Roman { 2 } -dominating function is the sum ? v ? V f ( v ) , and the minimum weight of a Roman { 2 } -dominating function f is the Roman { 2 } -domination number. First, we present bounds relating the Roman { 2 } -domination number to some other domination parameters. In particular, we show that the Roman { 2 } -domination number is bounded above by the 2-rainbow domination number. Moreover, we prove that equality between these two parameters holds for trees and cactus graphs with no even cycles. Finally, we show that associated decision problem for Roman { 2 } -domination is NP-complete, even for bipartite graphs.

On ve-degrees and ev-degrees in graphs

Abstract Let G = ( V , E ) be a graph with vertex set V and edge set E . A vertex v ∈ V v e -dominates every edge incident to it as well as every edge adjacent to these incident edges. The vertex–edge degree of a vertex v is the number of edges v e -dominated by v . Similarly, an edge e = u v e v -dominates the two vertices u and v incident to it, as well as every vertex adjacent to u or v . The edge–vertex degree of an edge e is the number of vertices e v -dominated by edge e . In this paper we introduce these types of degrees and study their properties.

A maximum degree theorem for diameter-2-critical graphs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n2/4⌋ and that the extremal graphs are the complete bipartite graphs K⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n0 where n0 is a tower of 2’s of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

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.

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.

Restricted optimal pebbling and domination in graphs

Abstract For a graph G = ( V , E ) , we consider placing a variable number of pebbles on the vertices of V . A pebbling move consists of deleting two pebbles from a vertex u ∈ V and placing one pebble on a vertex v adjacent to u . We seek an initial placement of a minimum total number of pebbles on the vertices in V , so that no vertex receives more than some positive integer t pebbles and for any given vertex v ∈ V , it is possible, by a sequence of pebbling moves, to move at least one pebble to v . We relate this minimum number of pebbles to several other well-studied parameters of a graph G , including the domination number, the optimal pebbling number, and the Roman domination number of G .

All My Favorite Conjectures Are Critical

My favorite graph theory conjectures involve the effects of edge removal on the diameter of a graph and the effects of edge addition on the domination and total domination numbers of a graph. Loosely speaking, “criticality” means that the value of the parameter in question always changes under the graph modification. This chapter presents five conjectures concerning criticality, namely, a conjecture by Sumner and Blitch on the criticality of domination upon edge addition, a conjecture by Murty and Simon on the criticality of diameter upon edge removal, and three conjectures on the criticality of total domination upon edge addition. These last three conjectures involving total domination are closely related, and surprisingly, a solution to one of them would provide a solution to the Murty-Simon Conjecture on diameter.