Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E′ ⊆ E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of E′ then E′ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to de...

In a (1 : b) Maker-Breaker game, one of the central questions is to find the maximal value of b that allows Maker to win the game (that is, the critical bias b∗). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of Kn is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, “Erdős Paradigm” turned out to be true. There...

The F&B-index is used to speed up pattern matching in tree and graph data, and is based on the maximum F&B-bisimulation, which can be computed in loglinear time for graphs. It has been shown that the maximum F-bisimulation can be computed in linear time for DAGs. We build on this result, and introduce a linear algorithm for computing the maximum F&B-bisimulation for tree data. It first computes...

Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n(11+o(1)) √ , significantly improving the previous best running time of O(2nn2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O( √ ...

Abstract. The F&B-index is used to speed up pattern matching in tree and graph data, and is based on the maximum F&B-bisimulation, which can be computed in loglinear time for graphs. It has been shown that the maximum F-bisimulation can be computed in linear time for DAGs. We build on this result, and introduce a linear algorithm for computing the maximum F&B-bisimulation for tree data. It firs...

A dominating set of an oriented graph D is a set S of vertices of D such that every vertex not in S is a successor of some vertex of S. The minimum cardinality of a dominating set of D, denoted γ(D), is the domination number of D. An irredundant set of an oriented graph D is a set S of vertices of D such that every vertex of S has a private successor, that is, for all x ∈ S, |O[x]− O[S − x]| ≥ ...

We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of edges of Gwhose deletion results in a subgraph with a unique perfect matching M , denoted by af (G,M). The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite ...

Let B(2m,m) be the set of all bicyclic graphs on 2m(m ≥ 2) vertices with perfect matchings. In this paper, we characterize the bicyclic graphs with minimal number of matchings and maximal number of independent sets in B(2m,m). 2010 Mathematics Subject Classification: 05C69, 05C05

Double hexagonal chains” can be considered as benzenoids constructed by successive fusions of successive naphthalenes along a zig-zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. In this paper, we discuss the numbers of k-matchings and k-independent sets of double hexagonal chains, as well as Hosoya indices and Merrifield–Simmons indices, and obtain some ex...

A well-known formula of Tutte and Berge expresses the size of a maximum matching in G in terms of what is usually called the deficiency of G. A subset X of G for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this paper, we study the structural aspects of maximal Tutte sets in a graph...