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The dynamics of the line and path graph operators

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Abstract

For any integerk e 1 thek- path graph Pk (G) of a graph G has all length-k subpaths ofG as vertices, and two such vertices are adjacent whenever their union (as subgraphs ofG) forms a path or cycle withk + 1 edges. Fork = 1 we get the well-known line graphP 1 (G) =L(G). Iteratedk-path graphs Pt k(G) are defined as usual by Pt k (G) := Pk(P t−1 k(G)) ift < 1, and by P1 k(G): = Pk(G). A graph G isP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -periodic. A graph has infiniteP k -depth if for any positive integert there is a graphH such that Pt k(H) ≃G. In this paperP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -periodic graphs,P k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -convergent graphs, and graphs with infiniteP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -depth are characterized inside some subclasses of the class of locally finite graphs fork = 1, 2.

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Correspondence to Erich Prisner.

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Prisner, E. The dynamics of the line and path graph operators. Graphs and Combinatorics 9, 335–352 (1993). https://doi.org/10.1007/BF02988321

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Key words

  • line graph
  • path graph
  • infinite graphs
  • graph dynamics