An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating diffi...

We address the problem of computing a Minimal Dominating Set in highly dynamic distributed systems. We assume weak connectivity, i.e., the network may be disconnected at each time instant and topological changes are unpredictable. We make only weak assumptions on the communication: every process is infinitely often able to communicate with other processes (not necessarily directly). Our contrib...

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

Many graph problems cannot be computed in polynomial time unless P = NP , which most computer scientists and mathematicians doubt. Examples are the Traveling Salesman Problem (TSP), vertex cover, and dominating set. TSP is to find a Hamiltonian cycle with the least weight in a complete weighted graph. A vertex cover of a graph is a subset of the vertices that covers all edges. A dominating set ...

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 review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D ...

In this paper we propose a new self-stabilizing distributed algorithm for minimal domination protocol in an arbitrary network graph using the synchronous model; the proposed protocol is general in the sense that it can stabilize with every possible minimal dominating set of the graph.

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper domination numbers, in symbols: Γ(G H) ≥ Γ(G)Γ(H). A conjecture posed by Vizing [7] in 1968 claims that Vizing’s conjecture: For any graphs G and H, γ(G H) ≥ γ(G)γ(H), where γ, as usual, denotes the do...

A hypergraph is a pair pV, Eq where V is a finite set and E Ď 2 is called the set of hyper-edges. An output-polynomial algorithm for C Ď 2 is an algorithm that lists without repetitions all the elements of C in time polynomial in the sum of the size of H and the accumulated size of all the elements in C. Whether there exists an output-polynomial algorithm to list all the inclusion-wise minimal ...