For graphs G and G' with minimum degree is = 3 and satisfying one of two other conditions, we prove that any isomorphism from the P4 -graph P4 (G) to P4 ( G') can be induced by a vertex-isomorphism of G onto G'. We also prove that a connected graph G is isomorphic to its P4 -graph P4 (G) if and only if G is a cycle of length at least 4.

The colored graph isomorphism problem is a restricted version of the general graph isomorphism (GI) problem that involves deciding the existence of a color preserving isomorphism between a pair of colored graphs. In this report, we study this problem for graphs whose color multiplicity is bounded by 3 (3-GI). We begin by formally defining the colored graph isomorphism problem and setup our nota...

A method for improving the efficiency of graph isomorphism testing is presented. The method uses the structure of the graph colored by vertex hash codes as a means of partitioning vertices into equivalence classes, which in turn reduces the combinatorial burden of isomorphism testing. Unrolling the graph into a tree at each vertex allows structurally different regular graphs to be discriminated...

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP. This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these problems share, as candidate NP-intermediate problems.

In Section 4.5, “Assertion 4-2 based on Theorem 1, 2, 3, 4: Two undirected and simple graphs G1 G2. The number of nodes for the two is same as n. edges m. There are no isolated in both graphs. vertex edge adjacency matrices generated V1 V2, E1 E2. sums rows/columns produce four sets an array aV1, bV2, aE1, bE1 If only if ∑n k=1 ak = bk (ak V1)2 (bk V2)2, then aV1 bV2 a permutation another set; ...

Abstract In the ( G , H )-isomorphism game, a verifier interacts with two non-communicating players (called provers), by privately sending each of them random vertex from either or . The goal is to convince that graphs and are isomorphic. recent work along Atserias et al. (J Comb Theory Ser B 136:89–328, 2019) we showed can be convinced non-isomorphic isomorphic, if provers allowed share quantu...

We study the equivalence problem of cubic forms. We lower bound its complexity by that of F-algebra isomorphism problem and hence by the graph isomorphism problem (for all fields F). For finite fields we upper bound the complexity of cubic forms by NP∩coAM. We also study the cubic forms obtained from F-algebras and show that they are regular and indecomposable.

We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard...

We present in this short note a polynomial graph extension procedure that can be used to improve any graph isomorphism algorithm. This construction propagates new constraints from the isomorphism constraints of the input graphs (denoted by G(V,E) and G(V , E)). Thus, information from the edge structures of G and G is ”hashed” into the weighted edges of the extended graphs. A bijective mapping i...

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other side, the relative complexity of the GI probl...