# Histories in path graphs

• Volume: 27, Issue: 2, page 345-357
• ISSN: 2083-5892

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## Abstract

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For a given graph G and a positive integer r the r-path graph, ${P}_{r}\left(G\right)$, 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 ${P}_{r}^{k}\left(G\right)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of ${P}_{r}^{k}\left(G\right)$. The k-history ${P}_{r}^{-k}\left(H\right)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.

## How to cite

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Ludovít Niepel. "Histories in path graphs." Discussiones Mathematicae Graph Theory 27.2 (2007): 345-357. <http://eudml.org/doc/270721>.

@article{LudovítNiepel2007,
abstract = {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 $P^k_r(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P^k_r(G)$. The k-history $P^\{-k\}_r(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.},
author = {Ludovít Niepel},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {path-graph; graph dynamics; history; graph operator; even caterpillar; convergent graphs; 1-2-bipartite graphs},
language = {eng},
number = {2},
pages = {345-357},
title = {Histories in path graphs},
url = {http://eudml.org/doc/270721},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Ludovít Niepel
TI - Histories in path graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2007
VL - 27
IS - 2
SP - 345
EP - 357
AB - 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 $P^k_r(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P^k_r(G)$. The k-history $P^{-k}_r(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.
LA - eng
KW - path-graph; graph dynamics; history; graph operator; even caterpillar; convergent graphs; 1-2-bipartite graphs
UR - http://eudml.org/doc/270721
ER -

## References

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