Abstract
A graph G is a B0-VPG graph if one can associate a horizontal or vertical path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. A graph G is a contactB0-VPG graph if it is a B0-VPG graph admitting a representation with no one-point paths, no two paths crossing, and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B0-VPG graphs within four special graph classes: chordal graphs, tree-cographs, P4-tidy graphs and P5-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact B0-VPG graphs.
Highlights
Golumbic et al introduced in [2] the concept of vertex intersection graphs of paths in a grid
An undirected graph G = (V, E) is called a VPG graph if one can associate a path in a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point
We considered some special graph classes, namely chordal graphs, tree-cographs, P4-tidy graphs and P5-free graphs
Summary
Golumbic et al introduced in [2] the concept of vertex intersection graphs of paths in a grid (referred to as VPG graphs). An undirected graph G = (V, E) is called a VPG graph if one can associate a path in a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. G = (V, E) is called a Bk-VPG graph, for some integer k ≥ 0, if one can associate a path with at most k bends in a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. In a recent paper (see [13]), contact Bk-VPG graphs have been investigated from a structural point of view and it was for instance shown that they do not contain cliques of size 7 and they always contain a vertex of degree at most 6. A preliminary version of this paper appears in [4]
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