Parameterized Complexity of Geodetic Set
A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb{N}$, the NP-hard ${\rm G{\small EODETIC}~S{ \small ET}}$ problem asks whether there is a geodetic set of size at most $k$. Complementing variou...
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| Published in: | Journal of graph algorithms and applications Vol. 26; no. 4; pp. 401 - 419 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
01.07.2022
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| ISSN: | 1526-1719, 1526-1719 |
| Online Access: | Get full text |
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| Summary: | A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb{N}$, the NP-hard ${\rm G{\small EODETIC}~S{ \small ET}}$ problem asks whether there is a geodetic set of size at most $k$.
Complementing various works on ${\rm G{\small EODETIC}~S{ \small ET}}$ restricted to special graph classes, we initiate a parameterized complexity study of ${\rm G{\small EODETIC}~S{ \small ET}}$ and show, on the one side, that ${\rm G{\small EODETIC}~S{ \small ET}}$ is $W[1]$-hard when parameterized by feedback vertex number, path-width, and solution size, combined.
On the other side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph. |
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| ISSN: | 1526-1719 1526-1719 |
| DOI: | 10.7155/jgaa.00601 |