A survey of Nordhaus-Gaddum type relations

Signed Domination in Block Graphs Signed Domination in Block Graphs

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number γS(D) of D. A set {f1, f2, . . . , fd} of signed dominating functions on D with the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (D), is called a signed dominating family (of functions) on D. The maximum number of functions in a signed dominating family on D is the signed domatic number of D, denoted by dS(D). In this work we show that 4 − n ≤ γS(D) ≤ n for each digraph D of order n ≥ 2, and we characterize the digraphs attending the lower bound as well as the upper bound. Furthermore, we prove that γS(D)+ dS(D) ≤ n+ 1 for any digraph D of order n, and we characterize the digraphs D with γS(D) + dS(D) = n+1. Some of our theorems imply well-known results on the signed domination number of graphs.

Signed domination and signed domatic numbers of digraphs

This glossary contains an annotated listing of some 300 parameters of graphs, together with their definitions, and, for most of these, a reference to the authors who introduced them. Let G = (V, E) be an undirected graph having order n = |V | vertices and size m = |E| edges. Two graphs G and H are isomorphic, denoted G ≃ H, if there exists a bijection ϕ : V (G) → V (H) such that two vertices u and v are adjacent in G if and only if the two vertices ϕ(u) and ϕ(v) are adjacent in H. For the purposes of this paper, we shall say that a parameter of a graph G is any integer-valued function \(f: \mathcal {G} \rightarrow \mathcal {Z}\) from the class of all finite graphs \(\mathcal {G}\) to the integers \(\mathcal {Z}\), such that for any two graphs G and H, if G is isomorphic to H then f(G) = f(H). This glossary also contains a listing of some 70 conjectures related to these parameters, more than 26 new parameters and open problem areas for study, and some 600 references to papers in which these parameters were introduced and then studied.

An Annotated Glossary of Graph Theory Parameters, with Conjectures

Signed star domatic number of a graph

Signed domatic number of a graph

Publisher Summary This chapter describes distances between isomorphism classes or distances between graphs. An isomorphism class of graphs is the class of all graphs that are isomorphic to a given graph. Two graphs whose distance is zero need not be identical but are isomorphic. A self-complementary graph is a graph that is isomorphic to its own complement. These graphs were studied independently by G. Ringel and H. Saclis. For the number n of vertices of a self-complementary graph, n 0 (mod 4) or n 1 (mod 4) always holds. An almost self-complementary graph can be defined as a graph that is isomorphic to a graph obtained from its complement by adding or deleting one edge.

Distances Between Graphs (Extended Abstract)

Publisher Summary This chapter presents some numerical invariants of graphs that are related to the concept of domination—namely, the domatic number and its variants.. The word domatic was coined from the words dominating and chromatic in the same way as the word smog was composed from the words smoke and fog. This concept is a certain analogy of the chromatic number, but instead of independent sets, dominating sets are used in its definition. A subset D of the vertex set V ( G ) of an undirected graphs G is called dominating if for each x V ( G ) − D there exists a vertex y D adjacent to x . A domatic partition of G is a partition of V ( G ), all of whose classes are dominating sets in G . The maximum number of classes of a domatic partition of G is called the “domatic number” of G and denoted by d ( G ). R. Laskar and S. T. Hedetniemi have introduced the connected domatic number d , ( G ) of a graph G . It is the maximum number of classes of a partition of V ( G ) into dominating sets that induce connected subgraphs of G .

Bohdan Zelinka

Papers

Signed domatic number of a graph

Distances Between Graphs (Extended Abstract)

Domatic Number of a Graph and its Variants (Extended Abstract)