We present a survey of con uence properties of (acyclic) term graph rewriting. Results and counterexamples are given for di erent kinds of term graph rewriting: besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations together are considered. Collapsing and copying together constitute bisimilarity of term graphs. We establis...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor $D$, while $D$ paired-dominating a~dominating and the subgraph induced by contains perfect matching. graph $D\!P\!D\!P$-graph it pair $(D,P)$ disjoint sets vertices such that $P$ $G$. The study $D\!P\!D\!P$-graphs was initiated Southey Henning (Cent. Eur. J. Math. 8 (2010) 459--467; Comb. Optim. 22 (2...

We show in this paper that the upper minus domination number −(G) of a claw-free cubic graph G is at most 1 2 |V (G)|. © 2006 Published by Elsevier B.V.

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n + O (n/g) < 3n/10 + O (n/g).

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...