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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.
We prove that for every graph H with the minimum degree 5, the third iterated line graph L3(H) of H contains K √ −1 as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number kG such that for every i kG the i-iterated line graph of G is 1 2 (L i(G))-linked. Since the degree of Li(G) is even, the result… (More)
A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words S is dominating if the sets S ∩N [u] where u ∈ V (G) and N [u] denotes the closed neighbourhood of u in G, are all nonempty. A set S ⊆ V (G) is called a locating code in G, if the sets S ∩ N [u] where u ∈ V (G) \ S are… (More)