On the enumeration of minimal non-pairwise compatibility graphs

All Graphs with at most 8 nodes are 2-interval-PCGs

“…We conclude by observing that 9 node graphs that are not PCGs are much more than seven. In fact, in [14] only minimal PCGs (i.e., each of their induced subgraphs is a PCG) with 9 nodes are enumerated and they are 1494. So, in order to prove (or disprove) that all 9 node graphs are 2-interval-PCGs, it does not seem possible to apply our approach, even resorting to a most efficient ILP model, as the ones described in [11,14].…”

“…More precisely, given a graph G and an unweighted tree T , the model determines (if they exist) an edgeweight function w for T , a bijective mapping σ between the node set V of G and the set of the leaves of T , and an interval I of the non-negative real halfline, such that G = P CG(T , I) where T is the edge-weighted tree obtained from T introducing weight w and mapping σ. After this result, in other papers [11,14] the ILP model has been made more efficient, in order to work on larger graphs. Since we handle few small graphs, here we adopt the first ILP model and opportunely modify it in order to let it work for recognizing whether a graph belongs to the wider class of 2-interval-PCGs.…”