Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, B k -EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B 4 -EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is B 3 -EPG, and that there exist circular-arc graphs which are not B 2 -EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle E-mail addresses: of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define B k -EPR graphs, k ≥ 0, the same way as B k -EPG graphs. Circulararc graphs are clearly B 4 -EPR graphs and we will show that there exist circular-arc graphs that are not B 3 -EPR graphs. We also show that normal circular-arc graphs are B 2 -EPR graphs and that there exist normal circular-arc graphs that are not B 1 -EPR graphs. Finally, we characterize B 1 -EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

An (h, s, t)-representation of a graph G consists of a collection of subtrees of a tree T , where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T . The class of graphs that has an (h, s, t)-representation is denoted by [h, s, t].An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h, 2, 1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h, 2, 1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h, 2, 1] is NP-complete, while the problem of deciding whether the graph belongs to [h, 2, 1] − [h − 1, 2, 1] is NP-hard. Both problems remain hard even when restricted to VPT ∩ Split. Additionally, we present a non-trivial subclass of VPT ∩ Split in which these problems are polynomial time solvable.

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called B 0 -VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time algorithm for recognizing chordal B 0 -VPG graphs. In this paper, we present a minimal forbidden induced subgraph characterization of B 0 -VPG graphs restricted to block graphs. As a byproduct, the proof of the main theorem provides an alternative certifying recognition and representation algorithm for B 0 -VPG graphs in the class of block graphs.

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