Related papers: Effectiveness and strong graph indivisibility

The graphical notion of effective resistance has found wide-ranging applications in many areas of pure mathematics, applied mathematics and control theory. By the nature of its construction, effective resistance can only be computed in…

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of…

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chv\'atal, stating that there exists…

A graph $G$ has a perfect division if its vertex set can be partitioned into two sets $A$, $B$ such that $G[A]$ is perfect and $\omega(G[B]) < \omega(G)$. We call $G$ perfectly divisible if every induced subgraph of $G$ admits a perfect…

We characterize when a subfactor $N\subseteq M$ is oracle computable relative to a presentation of the ambient factor $M$ in terms of computability of the Jones basic construction, in terms of computable Pismner-Popa bases, and in terms of…

Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions…

It is shown that a trace invariant projection map, i.e. a positive unital idempotent map, of a finite dimensional C*-algebra into itself is non-decomposable if and only if it is atomic, or equivalently not the sum of a 2-positive and a…

A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $\mathbb R$-linear. On the opposite direction of this result, we introduce a…

A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show…

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set…

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that when a quantum network is described by such a graph, the network is efficiently controllable, in…

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected or not and prove its correctness. The worst case run time complexity of the algorithm is shown to be exponential but still much better than…

An independent set may not contain both a vertex and one of its neighbours. This basic fact makes the uniform distribution over independent sets rather special. We consider the hard-core model, an essential generalization of the uniform…

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of…

The codegree of an irreducible character $\chi$ of a finite group $G$ is defined as $|G:\ker\chi|/\chi(1)$. The codegree graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct…

Full Bayesian computational inference for model determination in undirected graphical models is currently restricted to decomposable graphs, except for problems of very small scale. In this paper we develop new, more efficient methodology…

Given integers $\Delta\ge 2$ and $t\ge 2\Delta$, suppose there is a graph of maximum degree $\Delta$ and a partition of its vertices into blocks of size at least $t$. By a seminal result of Haxell, there must be some independent set of the…

A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its countable ultrapower over a cohesive set of natural numbers. A cohesive set is an…

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence…

Let A be a C*-algebra with real rank zero which has the stable weak cancellation property. Let I be an ideal of A such that I is stable and satisfies the corona factorization property. We prove that 0->I->A->A/I->0 is a full extension if…