Graph Isomorphism, Color Refinement, and Compactness

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomor...

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Veröffentlicht in:Computational complexity Jg. 26; H. 3; S. 627 - 685
Hauptverfasser: Arvind, V., Köbler, Johannes, Rattan, Gaurav, Verbitsky, Oleg
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.09.2017
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Automorphisms
Color
Combinatorial analysis
Computational Mathematics and Numerical Analysis
Computer Science
Graph theory
Graphs
Isomorphism
Linear programming
color refinement
linear programming relaxation
Graph Isomorphism
68Q25 Analysis of algorithms and problem complexity
polytope of fractional automorphisms
ISSN:1016-3328, 1420-8954
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Zusammenfassung:Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H . Babai et al. (SIAM J Comput 9(3):628–635, 1980 ) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time O ( ( n + m ) log n ) , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs . A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991 ) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: We show that all amenable graphs 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. 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.
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ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-016-0147-6