Abstract
Exploring a linear programming approach to Graph Isomorphism, Tinhofer (1991) defined the notion of compact graphs: A graph is compact if the polytope of its fractional automorphisms is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. In this paper we make new progress in our understanding of compact graphs. Our results are summarized below:
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We show that all graphs G which are distinguishable from any non-isomorphic graph by the classical color-refinement procedure are compact. In other words, the applicability range for Tinhofer’s linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement.
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Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.
This work was supported by the Alexander von Humboldt Foundation in its research group linkage program. The second author and the fourth author were supported by DFG grants KO 1053/7-2 and VE 652/1-2, respectively.
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The proof of Theorem 3 uses only compactness of complete graphs, matching graphs, and the 5-cycle.
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Arvind, V., Köbler, J., Rattan, G., Verbitsky, O. (2015). On Tinhofer’s Linear Programming Approach to Isomorphism Testing. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_3
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