On graphs that are not PCGs

A survey on pairwise compatibility graphs

“…Durocher et al showed a graph of eight vertices, as illustrated in Figure 4(b), and a planar graph of sixteen vertices which are not a PCG [16]. The following lemma illustrates a Calamoneri et al showed that the graph with eight vertices, which is not a PCG, is a circular arc graph, disk graph and rectangle intersection graph.…”

“…Given a graph, the PCG recognition problem asks to decide whether the graph is a PCG or not. Durocher et al considered a generalized PCG recognition problem and proved the hardness of the problem [16]. Given a graph G ¼ ðV, EÞ and a set S E, the generalized PCG recognition problem asks to find a PCG G 0 ¼ PCGðT, d min , d max Þ which contains G as a subgraph but does not contain any edge of S. If the maximum number of edges of S are required to have distance between their corresponding leaves greater than d max , then the generalized PCG recognition problem is NP-hard.…”

“…Durocher et al proved that the Max-Generalized PCG Recognition Problem is NP-hard [16] by reduction from the Monotone-One-In-Three-3-Sat problem [31]. However the complexity of the PCG recognition problem is still unknown [16].…”

“…A graph G a k-interval PCG if there is an edge weighted tree T for mutually exclusive intervals I 1 , I 2 , Á Á Á , I k of nonnegative real numbers such that each vertex of G corresponds to a leaf of T and there is an edge between two vertices in G if the distance between their corresponding leaves lies in I 1 [ I 2 [ Á Á Á I k [1]. Figure 7(b) illustrates an edge weighted tree T and Figure 7(a) shows the corresponding 2-interval PCG where I 1 ¼ ½1, 3 and I 2 ¼ ½5, 6: Note that the graph shown in Figure 7(b) is not a PCG [16].…”

“…constraint on the distances between pair of leaves based on which the graph with 8 vertices is proved to be not a PCG[16].Lemma 10. Let C be the cycle a 0 , b 0 , c 0 , d 0 of four vertices.…”