Graph Isomorphism, Color Refinement, and Compactness

Abstract

Abstract. 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 nonisomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined the notion of compact graphs: A graph is compact if its fractional automorphisms polytope 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. Our results are summarized below: -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 show that all amenable graphs are compact. Thus, 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.

Graphs Identified by Logics with Counting

“…Independently of our work, Arvind, Köbler, Rattan and Verbitsky [1] have investigated the structure of undirected graphs identified by C 2 obtaining results similar to the ones we provide in Section 4.…”

On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties

“…• Fürer [19] proved that C s ∈ C(2) for 3 ≤ s ≤ 6 and C s / ∈ C(2) for 8 ≤ s ≤ 16. We close the gap and show that C 7 is the largest cycle graph in C (2). We also prove that C(2) contains P 1 , .…”

Improving the Expressive Power of Graph Neural Network with Tinhofer Algorithm

“…It is also known that a graph that can be distinguished from any non-isomorphic graph by WL test is compact. More compact graphs and the relation between compactness, graph isomorphism, and WL test can be found in [1,2].…”

“…However, it is also a compact graph that could be identified by Tinhofer algorithm. Moreover, it is proved in [2] that if WL test could distinguish a graph G from any non-isomorphic graph H, then the graph G is compact.…”