The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)...

We motivate and discuss four open problems in polyhedral combinatorics related to threshold graphs, degree sequences of graphs and hypergraphs, and balanced subgraphs of 2-colored graphs.

A graph is balanced if its clique-matrix contains no edge-vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs ...

Given a graph G with n vertices and k players, each of which is placing a facility on one of the vertices of G, we define the score of the ith player to be the number of vertices for which, among all players, the facility placed by the ith player is the closest. A placement is balanced if all players get roughly the same score. A graph is balanced if all placements on it are balanced. Viewing b...

A biased graph consists of a graph T and a subclass B of the polygons of T, such that no theta subgraph of T contains exactly two members of B. A subgraph is balanced when all its polygons belong to B. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph ha...

A graph G is strongly distance-balanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v)−1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distancebalanced. It is also proved that connected components of the direct pro...