We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we ch...

Semiorders may form the simplest class of ordered sets with a not necessarily transitive indifference relation. Their generalization has given birth to many other classes of ordered sets, each of them characterized by an interval representation, by the properties of its relations or by forbidden configurations. In this paper, we are interested in preference structures having an interval represe...

A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe int...

Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a lineartime algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any matrix that does not have the co...

Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a lineartime algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any matrix that does not have the co...

A poset P = (X, 4 ) is a split semiorder if there are maps a , f : X --* • with a(x)<~f(x)<~a(x) + 1 for every x E X such that x -< y if and only if f ( x ) < a ( y ) and a(x) + 1 < f ( y ) . A split interval order is defined similarly with a(x)+ 1 replaced by b(x), a(x)<~ f(x)<~ b(x), such that x -< y if and only if f ( x ) < a ( y ) and b(x) < f ( y ) . We investigate these generalizations of...

The Gram dimension gd(G) of a graph G is the smallest integer k ≥ 1 such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists). For any fixed k the class of graphs satisfying gd(G) ≤...

A cycle-transversal of a graph G is a subset T ⊆ V (G) such that T ∩V (C) 6= ∅ for every cycle C of G. A clique cycle-transversal, or cct for short, is a cycle-transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distancehereditary graphs admitting a cct in terms of fo...

We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.