Long path connectivity of regular graphs

Long path connectivity of regular graphs

Connectivity of path graphs

We prove a necessary and sufficient condition under which a connected graph has a connected P3-path graph. Moreover, an analogous condition for connectivity of the Pk-path graph of a connected graph which does not contain a cycle of length smaller than k+1 is derived.

Three-regular path pairable graphs

A graph G with at least 2k vertices is k-path pairable if for any k pairs of distinct vertices 9f G there are k edge disjoint paths between the pairs. It will be shown for any positive integer k that there is a k-path pairable graph of maximum degree three.

Path Algorithms on Regular Graphs

We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars.

Restricted Edge Connectivity of Regular Graphs

An edge cut that separates the connected graph into components with order at least two is restricted edge cut. The cardinality of minimum restricted edge cut is restricted edge connectivity. Denote byλ′ ( G) the restricted edge connectivity , then λ′( G) ≤ξ( G) where ξ( G) is the minimum edge degree. G is called maximal restricted edge connected if the equality in the previous inequality holds....