Related papers: Computing fixed point free automorphisms of graphs

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if…

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many…

Given a class of graphs $\mathcal{H}$, the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is defined as follows. The input is a graph $H\in \mathcal{H}$ together with an arbitrary graph $G$. The problem is to compute, modulo $2$, the number of…

Cographs are graphs in which no four vertices induce a simple connected path $P_4$. Cograph editing is to find for a given graph $G = (V,E)$ a set of at most $k$ edge additions and deletions that transform $G$ into a cograph. This…

We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter…

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this…

We consider the problem of finding a subgraph of a given graph minimizing the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already for bipartite graphs when the functions are convex on…

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for…

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general…

We study the automorphisms \phi of a finitely generated free group F. Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative f:G\to G of a power of \phi that behaves very much like the…

Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in…

We prove that Graph Isomorphism and Canonization in graphs excluding a fixed graph $H$ as a minor can be solved by an algorithm working in time $f(H)\cdot n^{O(1)}$, where $f$ is some function. In other words, we show that these problems…

Let $\sigma$ be an automorphism of a field $K$ with fixed field $F$. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras $K[t;\sigma]/fK[t;\sigma]$ obtained…

A claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph, and a 2-factor is a 2-regular spanning subgraph of a graph. In 1997, Ryj\'{a}\v{c}ek introduced the closure concept of claw-free graphs, and Hamilton…

For a fixed graph $F$, let $ex_F(G)$ denote the size of the largest $F$-free subgraph of $G$. Computing or estimating $ex_F(G)$ for various pairs $F,G$ is one of the central problems in extremal combinatorics. It is thus natural to ask how…

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge…

Over 50 years ago, Lov\'{a}sz proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph [Acta Math. Hungar. 18 (1967), pp. 321--328]. In this work we prove that two graphs are quantum…

Given a finite set of $2$-edge-coloured graphs $\mathcal F$ and a hereditary property of graphs $\mathcal{P}$, we say that $\mathcal F$ expresses $\mathcal{P}$ if a graph $G$ has the property $\mathcal{P}$ if and only if it admits a…

For a graph property $\Pi$, Subgraph Complementation to $\Pi$ is the problem to find whether there is a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph…