The Upper Domatic Number of a Graph

Let (Formula presented.) be a graph. For two disjoint sets of vertices (Formula presented.) and (Formula presented.), set (Formula presented.) dominates set (Formula presented.) if every vertex in (Formula presented.) is adjacent to at least one vertex in (Formula presented.). In this paper we introduce the upper domatic number (Formula presented.), which equals the maximum order (Formula presented.) of a vertex partition (Formula presented.) such that for every (Formula presented.), (Formula presented.), either (Formula presented.) dominates (Formula presented.) or (Formula presented.) dominates (Formula presented.), or both. We study properties of the upper domatic number of a graph, determine bounds on (Formula presented.), and compare (Formula presented.) to a related parameter, the transitivity (Formula presented.) of (Formula presented.)

The 2-dimension of a tree

Let x and y be two distinct vertices in a connected graph G. The x, ylocation of a vertex w is

Self-coalition graphs

A coalition in a graph $G = (V, E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 \cup V_2$ is a dominating set of $G$. A coalition partition in a graph $G$ of order $n = |V|$ is a vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ such that every set $V_i$ either is a dominating set consisting of a single vertex of degree $n-1$, or is not a dominating set but forms a coalition with another set $V_j$ which is not a dominating set. Associated with every coalition partition $\pi$ of a graph $G$ is a graph called the coalition graph of $G$ with respect to $\pi$, denoted $CG(G,\pi)$, the vertices of which correspond one-to-one with the sets $V_1, V_2, \ldots, V_k$ of $\pi$ and two vertices are adjacent in $CG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a coalition. The singleton partition $\pi_1$ of the vertex set of $G$ is a partition of order $|V|$, that is, each vertex of $G$ is in a singleton set of the partition. A graph $G$ is called a self-coalition graph if $G$ is isomorphic to its coalition graph $CG(G,\pi_1)$, where $\pi_1$ is the singleton partition of $G$. In this paper, we characterize self-coalition graphs

On Unique Minimum Dominating Sets in Some Cartesian Product Graphs

Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set

On Unique Minimum Dominating Sets in Some Cartesian Product Graphs

Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set

The Transitivity of Special Graph Classes

Let G = (V,E) be a graph. The transitivity of a graph G, denoted Tr(G), equals the maximum order k of a partition x = {Vi,of V such that for every i,j, 1 i j k, Vi dominates Vj. We consider the transitivity in many special classes of graphs, including cactus graphs, coronas, Cartesian products, and joins. We also consider the effects of vertex or edge deletion and edge addition on the transivity of a graph

Introduction to Coalitions in Graphs

A coalition in a graph (Formula presented.) consists of two disjoint sets of vertices V 1 and V 2, neither of which is a dominating set but whose union (Formula presented.) is a dominating set. A coalition partition in a graph G of order (Formula presented.) is a vertex partition (Formula presented.) such that every set Vi of π either is a dominating set consisting of a single vertex of degree n–1, or is not a dominating set but forms a coalition with another set (Formula presented.) which is not a dominating set. In this paper we introduce this concept and study its properties

Coalition Graphs of Paths, Cycles, and Trees

A coalition in a graph G =(V, E) consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set of G but whose union V1 ∪ V2 is a dominating set of G.A coalition partition in a graph G of order n = |V| is a vertex partition π= {V1, V2,⋯, Vk} of V such that every set Vi either is a dominating set consisting of a single vertex of degree n - 1, or is not a dominating set but forms a coalition with another set Vj which is not a dominating set. Associated with every coalition partition πof a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G, π), the vertices of which correspond one-to-one with the sets V1, V2,⋯, Vk of πand two vertices are adjacent in CG(G, π) if and only if their corresponding sets in πform a coalition. In this paper we study coalition graphs, focusing on the coalition graphs of paths, cycles, and trees. We show that there are only finitely many coalition graphs of paths and finitely many coalition graphs of cycles and we identify precisely what they are. On the other hand, we show that there are infinitely many coalition graphs of trees and characterize this family of graphs

Upper Bounds on the Coalition Number

A dominating set in a graph G = (V,E) is a set S ⊆ V such that every vertex not in S is adjacent to at least one vertex in S. A coalition in a graph G consists of two disjoint sets V1, V2 ⊂ V neither of which is a dominating set but whose union V1∪ V2 is a dominating set. A vertex partition π = {V1, V2, …, Vk} such that every set Vi is either a dominating set consisting of a single vertex, or is not a dominating set but forms a coalition with another set Vj which is not a dominating set, is called a coalition partition. The maximum order of a coalition partition is called the coalition number of G. In this paper we obtain a tight upper bound on the coalition number of any graph G in terms of the maximum degree of G. We also give a tight upper bound on the coalition number in terms of both maximum degree and minimum degree of G

Vertex-Edge Domination

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertexedge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number γve(G). We present bounds on γve(G) and relationships between γve(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that γve(G) is bounded above by the domination number of G. Our main result characterizes the trees having equal domination and vertex-edge domination numbers