A discriminatory theorem of Kuratowski subgraphs

Abstract

The definition of a critical non-planar graph with respect to one of its Tr~maux cotrees provides a new Kuratowski theorem. A straightforward consequence is a simple discriminatory criterion for the search in linear time of a subdivision of K3, 3 or K 5 in non-planar graphs.

On Cotree-Critical and DFS Cotree-Critical Graphs

“…Thus, no two pairs (x, y) and (z, y) may belong to S. Then, the set {{x, y} : (x, y) ∈ S or (y, x) ∈ S} is a matching of K 3,3 . As S includes at least 4 pairs and as K 3,3 has no matching of size greater than 3, we are led to a contradiction. Proof: If all the non-critical graphs belong to some simple path, the set of the non-critical edges is acyclic and the graph is cotree critical.…”

“…A Kuratowski subdivision in a graph G is a minimal non-planar subgraph of G, that is: a non-planar subgraph K of G, such that all the edges of K are critical for K. Kuratowski proved in [6] that such minimal graphs are either subdivisions of K 5 or subdivisions of K 3,3 .…”

“…Let S be the set of the pairs (x, y) of K-vertices, such that there exists a K-subvertex v adjacent to x belonging to a K-branch having y as one of its endpoints. Notice that K + {x, v} − {x, y} is a subdivision of K 3,3 and thus that [x, y] is non-critical for G.…”

“…cotree) edge is critical, in the sense that its deletion would lead to a planar graph. A first study of DFS cotree-critical graphs appeared in [3], in which it is proved that a DFS cotree-critical graph either is isomorphic to K 5 or includes a subdivision of K 3,3 and no subdivision of K 5 . The linear time Kuratowski subdivision extraction algorithm, which has been both conceived and implemented in [2] by the authors, consists in two steps: the first one correspond to the extraction of a DFS cotree-critical subgraph by a case analysis algorithm; the second one extracts a Kuratowski subdivision from the DFS cotree-critical subgraph by a very simple algorithm (see Algorithm 1), but which theoretical justification is quite complex and relies on the full characterization of DFS cotree-critical graphs that we prove in this paper: a simple graph is DFS cotree-critical if and only if it is either K 5 or a Möbius pseudo-ladder having a simple path including all the non-critical edges (see Figure 1).…”

On Cotree-Critical and DFS Cotree-Critical Graphs

“…Then, three pairwise non-adjacent non-critical edges are found to complete a Kuratowski subdivision of G isomorphic to K 3,3 .…”