A matching in a hypergraph H is a set of pairwise disjoint hyperedges. The matching number α (H) of H is the size of a maximum matching in H. A subset D of vertices of H is a dominating set of H if for every v ∈ V \ D there exists u ∈ D such that u and v lie in an hyperedge of H. The cardinality of a minimum dominating set of H is called the domination number of H, denoted by γ(H). It is known that for a intersecting hypergraph H with rank r, γ(H) ≤ r − 1. In this paper we present structural properties on intersecting hypergraphs with rank r satisfying the equality γ(H) = r − 1. By applying the properties we show that all linear intersecting hypergraphs H with rank 4 satisfying γ(H) = r − 1 can be constructed by the well-known Fano plane.
We consider transferable utility cooperative games (TU games) with limited cooperation introduced by hypergraph communication structure, the so-called hypergraph games. A hypergraph communication structure is given by a collection of coalitions, the hyperlinks of the hypergraph, for which it is assumed that only coalitions that are hyperlinks or connected unions of hyperlinks are able to cooperate and realize their worth. We introduce the average tree value for hypergraph games, which assigns to each player the average of the player's marginal contributions with respect to a particular collection of rooted spanning trees of the hypergraph. We also provide axiomatization of the average tree value for hypergraph games on the subclasses of cycle-free hypergraph games, hypertree games and cycle hypergraph games.
A function f : V (G) → {+1, 0, −1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least one. The minus total domination number γ − t (G) of G is the minimum weight of a minus total dominating function on G. By simply changing "{+1, 0 − 1}" in the above definition to "{+1, −1}", we can define the signed total dominating function and the signed total domination number γ s t (G) of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on γ s t and γ − t for triangle-free graphs and characterize the extremal graphs achieving these bounds.
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