Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are char...

Two new domination parameters for a connected graph G: the weakly convex domination number of G and the convex domination number of G are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

A mixed dominating set S of a graph G = (V,E) is a subset S ⊆ V ∪ E such that each element v ∈ (V ∪ E) \ S is adjacent or incident to at least one element in S. The mixed domination number γm(G) of a graph G is the minimum cardinality among all mixed dominating sets in G. The problem of finding γm(G) is know to be NP-complete. In this paper, we present an explicit polynomial-time algorithm to c...

In the state-based self-stabilizing algorithmic paradigm for distributed computing, each node has only a local view of the system (seeing its neighbors’ states), yet in a finite amount of time the system converges to a global state satisfying some desired property. In this paper we introduce a general mechanism that allows a node to act only on correct distance-two knowledge, provided there are...

Data reduction by polynomial-time preprocessing is a core concept of (parameterized) complexity analysis in solving NP-hard problems. Its practical usefulness is confirmed by experimental work. Here, generalizing and extending previous work, we present a set of data reduction preprocessing rules on the way to compute optimal dominating sets in graphs. In this way, we arrive at the novel notion ...

Let G = (V,E) be a connected graph. A connected dominating set S ⊂ V is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted γc(G), is the minimum cardinality of a connected dominating set. Alternatively, |V | − γc(G) is the maximum number of leaves in a spanning tree of G. Let δ denote the minimum degree of G. We prove that γc(G) ≤ |V | ln(δ+1)...

In recent years, finding new satisfiability algorithms for various circuit classes has been a very active line of research. Despite considerable progress, we are still far away from a definite answer on which circuit classes allow fast satisfiability algorithms. This survey takes a (far from exhaustive) look at some recent satisfiability algorithms for a range of circuit classes and highlights ...

We establish a broad generalization of Whitney’s broken circuit theorem on the chromatic polynomial of a graph to sums of the type ∑ A⊆S f(A) where S is a finite set and f is a mapping from the power set of S to an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polyno...

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...