### On the Relationship Between Map Graphs and Clique Planar Graphs [chapter]

Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, Ignaz Rutter
2015 Lecture Notes in Computer Science
A map graph is a contact graph of internally-disjoint regions of the plane, where the contact can be even a point. Namely, each vertex is represented by a simple connected region and two vertices are connected by an edge iff the corresponding regions touch. Map graphs are introduced in [2] to allow the representation of graphs containing large cliques in a readable way. A clique planar graph is a graph G = (V, E) that admits a representation where each vertex u ∈ V is represented by an
more » ... llel unit square R(u) and where, for some partition of V into vertex-disjoint cliques S = {c 1 , . . . , c k }, each edge (u, v) is represented by the intersection between R(u) and R(v) if u and v belong to the same clique (intersection edges) or by a non-intersected curve connecting the boundaries of R(u) and R(v) otherwise (link edges); see Fig. 1(c) . Clique planar graphs are introduced in [1], where it is mainly addressed the case where the clique partition S is given. Figure 2 provides an example of a graph that is both a map graph and a clique planar graph. In [1] it is argued that there are graphs that admit a cliqueplanar representation while not admitting any representation as a map graph, and vice versa. In this poster we exhibit such counterexamples, establishing that neither of the classes of map graphs and of clique planar graphs is contained in the other. Lemma 1. There exists a clique planar graph that is not a map graph. Proof. Consider the graph G of Fig. 2(a) . Observe that G is not planar since vertices 1, 3, 4, 5, and 6 (filled red in Fig. 2(a) ) form a K 5 subdvision. However, graph G is clique planar (see Fig. 2(b) ). If G were also a map graph, some edges could be represented by regions sharing a point. Since only the two triangles 1, 2, 3 and 4, 5, 6 could be represented in such a way, this would imply the pla-