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MATHEMATICA BOHEMICA, Vol. 127, No. 3, pp. 473-480 (2002)
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# Graphs isomorphic to their path graphs

## Martin Knor, Ludovit Niepel

* Martin Knor*, Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics, Radlinského 11, 813 68 Bratislava, Slovakia, e-mail: ` knor@vox.svf.stuba.sk`

* Ludovit Niepel*, Kuwait University, Faculty of Science, Department of Mathematics & Computer Science, P.O. Box 5969, Safat 13060, Kuwait, e-mail: ` niepel@math-1.sci.kuniv.edu.kw`

**Abstract:**
We prove that for every number $n\ge 1$, 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 $k\ge 4$ we reduce the problem of characterizing graphs $G$ such that $P_k(G)\cong G$ to graphs without cycles of length exceeding $k$.

**Keywords:** line graph, path graph, cycles

**Classification (MSC2000):** 05C38

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