In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp{Õ(n 13 )} (this bound was recently improved to exp{Õ(n 15 )} [5]). It has since been an open question to remove the requirement that the graph...

In this paper we focus on connected directed/undirected circulant graphs Cn(a, b). We investigate some topological characteristics, and define a simple combinatorial model, which is new for the topic. Building on such a model, we derive a necessary and sufficient condition to test whether two circulant graphs Cn(a, b) and Cn(a , b) are isomorphic or not. The method is entirely elementary and co...

Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set...

In this paper we introduce and investigate a new class of graphs called algebraic forests for which isomorphism testing can be done in time O(n3 log n). The class of algebraic forests admits a membership test of the same complexity, it includes cographs, trees and interval graphs, and even a joint superclass of the latter two, namely, rooted directed path graphs. In fact, our class is much larg...

We study the problem of processing supergraph queries on graph databases. A graph database D is a large set of graphs. A supergraph query q on D is to retrieve all the graphs in D such that q is a supergraph of them. The large number of graphs in databases and the NP-completeness of subgraph isomorphism testing make it challenging to efficiently processing supergraph queries. In this paper, a n...

In this paper we present an algorithm for testing the isomorphism of two graphs. The algorithm works in three steps. First, it builds a sequence of partitions on the vertices of one of the graphs. Then, it looks for some automorphisms in that graph. Finally, it uses backtracking to try to find another sequence of partitions for the second graph that is compatible with that for the first graph. ...

While in general it is not known whether there is a polynomial time algorithm to decide whether two given graphs are isomorphic, there are polynomial-time algorithms for certain subsets of graphs, including but not limited to planar graphs and graphs with bounded valence. In this thesis, we will give a brief introduction on the Graph Isomorphism Problem and its relations to complexity theory. W...

We define two polynomials f(G) and /*(G) for a graph G by a recursive formula with respect to deformation of graphs. Analyzing their various properties, we shall discuss when two graphs have the same polynomials. Introduction. Our graphs are finite, undirected ones with or without self-loops and multiple edges. As elementary deformations of graphs, we have the deletion and the contraction of ed...

The class of problems solvable in logarithmic space has recently replenished with the isomorphism testing for interval graphs. We discuss this result, prospects of its extension to larger classes of graphs, and related issues such as constructing canonical models of intersection graphs and solving the Star System Problem for restricted classes of graphs.

Let C be a class of graphs closed under isomorphism and let ~ C be obtained from C by arc anchorage. A concept of ~ C-homogeneous graphs that include the C-ultrahomogeneous graphs is given, with the explicit construction of ~ C-homogeneous graphs that are not C-ultrahomogeneous, via ordered pencils of binary projective spaces.