We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (Vapnik-Chervonenkis) dimension of these set systems. For most of these set systems (e.g. for the systems induced by trees, connected sets, or paths), computing the VC-dimension is an NP-hard problem. Moreover, determining the VC-dimension for set systems induced by neighborhoods of single vertices is complete for the class LoGNP. In contrast to these intractability results, we show that the VC-dimension for set systems induced by stars is computable in polynomial time. For set systems induced by paths, we determine the extremal graphs G with the minimum number of edges such that VCP(G) ≥ k. Finally, we show a close relation between the VC-dimension of set systems induced by connected sets of vertices and the VC dimension of set systems induced by connected sets of edges; the argument is done via the line graph of the corresponding graph.
CITATION STYLE
Kranakis, E., Krizanc, D., Ruf, B., Urrutia, J., & Woeginger, G. J. (1995). VC-dimensions for graphs (extended abstract). In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1017, pp. 1–13). Springer Verlag. https://doi.org/10.1007/3-540-60618-1_61
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