Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1802.03611

Searching isomorphic graphs

Anatoly D. Plotnikov
View original source ↗Read source document ↗Export metadata ↓

Abstract

To determine that two given undirected graphs are isomorphic, we construct for them auxiliary graphs, using the breadth-first search. This makes capability to position vertices in each digraph with respect to each other. If the given graphs are isomorphic, in each of them we can find such positionally equivalent auxiliary digraphs that have the same mutual positioning of vertices. Obviously, if the given graphs are isomorphic, then such equivalent digraphs exist. Proceeding from the arrangement of vertices in one of the digraphs, we try to determine the corresponding vertices in another digraph. As a result we develop the algorithm for constructing a bijective mapping between vertices of the given graphs if they are isomorphic. The running time of the algorithm equal to $O(n^5)$, where $n$ is the number of graph vertices.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anatoly D. Plotnikov. 2018-02-10. Searching isomorphic graphs. https://doi.org/10.14738/tnc.55.3551

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Independent Set Reconfiguration via Dilworth Decompositions

The Token Jumping and Sliding Token problems are fundamental reconfiguration problems defined on the independent sets of an undirected graph. Given two independent sets $I$ and $J$, each of size $k$, these problems ask whether there exists a sequence of elementary operations transforming $I$ into $J$ such that every intermediate configuration is also an independent set of size $k$. Suppose a token is placed on each vertex of $I$: in Sliding Token, an operation moves a token from a vertex $u \in I$ to an adjacent vertex $v \notin I$; in Token Jumping, the token may instead move to any vertex $v \notin I$. While both problems are $\mathsf{PSPACE}$-complete on general graphs, polynomial-time algorithms for one or both variants have been developed for several graph classes, including trees, block graphs, bipartite permutation graphs, cographs, $P_4$-tidy graphs, and interval graphs. In this paper, we prove that both problems are solvable in polynomial time on threshold signed graphs, also known as Dilworth-2 graphs. A graph $G=(V,E)$ is a threshold signed graph if there exist a mapping $a:V\to\mathbb{R}$ and positive real constants $S,T>0$ such that $|a(v)|< \min\{S,T\}$ for all $v \in V$, and for any distinct vertices $u,v\in V$, $\{u,v\}\in E$ if and only if $|a(u)+a(v)|\ge S$ or $|a(u)-a(v)|\ge T$. More generally, we also show that Token Jumping can be solved in time $n^{O(\mathcal{D}(G))}$, where $\mathcal{D}(G)$ denotes the Dilworth number of $G$. Thus, Token Jumping belongs to $\mathsf{XP}$ when parameterised by the Dilworth number. This graph class is a subclass of permutation graphs, for which the complexity of these problems remains open, and is incomparable with the class of bipartite permutation graphs studied by Fox-Epstein et al. (ISAAC, 2015).

cs.DS↗
arXiv2026

Matrix Spencer: Eight Standard Deviations Suffice and an Almost-Linear Time Algorithm for Dense Input

The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We also give a randomized algorithm that finds a signing of discrepancy below $12\sqrt n$ with failure probability at most $p$. The algorithm uses $n^{3+o(1)}\operatorname{polylog}(1/p)$ arithmetic operations in the real-arithmetic model. This matches the size $n^3$ of the dense input up to subpolynomial factors. In the other direction, we prove that for every $n$ there are collections of symmetric matrices such that every signing has discrepancy at least $(2-o(1))\sqrt{n}$. We present three different proofs of the matrix Spencer conjecture. The key to every proof is a hereditary small-ball estimate. This is a lower bound on the Gaussian measure of the spectral body $\{x\in \mathbb{R}^n:\|\sum_ix_iA_i\|\le R\}$ that holds for every subfamily of the matrices. The other ingredient turns that Gaussian measure into a partial signing. We give three approaches to obtain such a signing. The first one covers the cube by partially signed faces through Gaussian concentration with a constant $7\cdot10^9$. The second proof replaces the covering by a projection lemma with explicit parameters for a constant $156000$. The third proof turns Gaussian measure into signs by a lossless coding, with no union bound. It proves the estimate at the right radius with smooth spectral barriers and certified coefficients. It gives a constant below $7.88$. For algorithms, the main idea is to project Gaussian points onto a smoothed spectral body. The $n^{3+o(1)}$ time algorithm tracks the Gibbs matrix of that body across coordinate-descent steps with sketched increments and random refreshes.

cs.DS↗
arXiv2026

On Deterministically Computing Total Variation Distance via Zonotope Compression

We study deterministic relative approximation of the total variation distance between high-dimensional distributions given by succinct descriptions. We develop an abstract deterministic approximation framework based on representing the total variation distance as a support function of a low-dimensional zonotope. As applications, we obtain FPTASs for several models. Given two mixtures of product distributions over $[q]^n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)^{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]^n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq^2(n/\varepsilon)^{2K})$. Finally, for two latent-tree Ising models with the same underlying tree topology, we give an FPTAS for the TV-distance between their leaf marginals in time $O(|V|^{13}\varepsilon^{-12})$.

cs.DS↗