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Histories in path graphs

Ludovít Niepel — 2007

Discussiones Mathematicae Graph Theory

For a given graph G and a positive integer r the r-path graph, $P_{r} (G)$ , has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_{r}^{k} (G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_{r}^{k} (G)$ . The k-history $P_{r}^{- k} (H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of...

On decompositions of complete graphs into factors with given diameters and radii

Ľudovít Niepel — 1980

Mathematica Slovaca

Elevation of a graph

Ľudovít Niepel; Pavel Tomasta — 1981

Czechoslovak Mathematical Journal

Graphs isomorphic to their path graphs

Martin Knor; Ľudovít Niepel — 2002

Mathematica Bohemica

We prove that for every number $n \geq 1$ , the $n$ -iterated $P_{3}$ -path graph of $G$ is isomorphic to $G$ if and only if $G$ is a collection of cycles, each of length at least 4. Hence, $G$ is isomorphic to $P_{3} (G)$ if and only if $G$ is a collection of cycles, each of length at least 4. Moreover, for $k \geq 4$ we reduce the problem of characterizing graphs $G$ such that $P_{k} (G) ≅ G$ to graphs without cycles of length exceeding $k$ .

Centers in line graphs

Martin Knor; Ľudovít Niepel; Ľubomír Šoltés — 1993