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- W2966575401 abstract "The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $kgeq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick cite{FiFi01} have shown that the isometric path cover number of $(rtimes r)$-dimensional grid is $lceil 2r/3rceil$. We show that the isometric path cover (partition) number of $(rtimes s)$-dimensional grid is $s$ when $r geq s(s-1)$. We establish that the isometric path cover (partition) number of $(rtimes r)$-dimensional torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then, we demonstrate that the isometric path cover (partition) number of an $r$-dimensional Benes network is $2^r$. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids." @default.
- W2966575401 created "2019-08-13" @default.
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- W2966575401 date "2021-01-01" @default.
- W2966575401 modified "2023-09-23" @default.
- W2966575401 title "On the isometric path partition problem" @default.
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- W2966575401 doi "https://doi.org/10.7151/dmgt.2236" @default.
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