Meshes on 3-connected graphs

On 3*-connected graphs

Menger's Theorem states that in a 3-connected graph, any two vertices are joined by three openly disjoint paths. Here we consider 3-connected cubic graphs where two vertices exist so that the three disjoint paths between them contain all of the vertices of the graph (we call these graphs 3*connected); and also where the latter is true for ALL pairs of vertices (globally 3*-connected). A necessa...

A note on minimally 3-connected graphs

If G is a minimally 3-connected graph and C is a double cover of the set of edges of G by irreducible walks, then |E (G)| ≥ 2 |C| − 2.

On 3-connected hamiltonian line graphs

Thomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and...

On contractible edges in 3-connected graphs

The existence of contractible edges is a very useful tool in graph theory. For 3-connected graphs with at least six vertices, Ota and Saito (1988) prove that the set of contractible edges cannot be covered by two vertices. Saito (1990) prove that if a three-element vertex set S covers all contractible edges of a 3-connected graph G, then S is a vertex-cut of G provided that G has at least eight...

Decomposition of 3-connected graphs