Graphs of edge-intersecting non-splitting paths in a tree: Representations of holes-Part I

Given a tree and a set of non-trivial simple paths on it, V-PT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T, and E-PT(P)) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices are the vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path).