A graph G is a max point-tolerance (MPT) graph if each vertex v of G can be mapped to a pointed-interval (I v , p v ) where I v is an interval of R and p v ∈ I v such that uv is an edge of G iff I u ∩ I v ⊇ {p u , p v }. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.

Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.-We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. -We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). -We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.

A hole is a chordless cycle with at least four vertices. A pan is a graph that consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our ( )-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our ( 2.5 + )-time algorithm to optimally color them.Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 time the clique number.

A graph is B k -VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n 3/2 ) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2-DIR.

a b s t r a c tIn this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: , , , , and we consider zero bend paths (i.e., | and -) to be degenerate 's. These graphs, called B 1 -EPG graphs, were first introduced by Golumbic et al. (2009). We consider the natural subclasses of B 1 -EPG formed by the subsets of the four single bend shapes (i.e., { }, { , }, { , }, and { , , }) and we denote the classes by [ ], [ , ], [ , ], and [ , , ] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [ ] [ , ], [ , ] [ , , ] B 1 -EPG and [ , ] is incomparable with [ , ]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩ [ ].

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Biró, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on H-graphs.We show that for any fixed H containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on H-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on H-graphs. Namely, when H is a cactus the clique problem can be solved in polynomial time. Also, when a graph G has a Helly H-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the k-clique and list k-coloring problems are FPT on H-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number.

In this paper we study properties of intersection graphs of kbend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B k -VPG. We show here that for every fixed k, B k -VPG B k+1 -VPG and that recognition of graphs from B k -VPG is NP-complete even when the input graph is given by a B k+1 -VPG representation. We also show that the class B k -VPG (for k ≥ 1) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.