Related papers: Local Orthogonality Dimension
We analyze the query complexity of finding a local minimum in $t$ rounds on general graphs. More precisely, given a graph $G = (V,E)$ and oracle access to an unknown function $f : V \to \mathbb{R}$, the goal is to find a local minimum--a…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes…
Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and vertex conductance $\Psi^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap…
Orthogonal graph drawings are used in applications such as UML diagrams, VLSI layout, cable plans, and metro maps. We focus on drawing planar graphs and assume that we are given an \emph{orthogonal representation} that describes the desired…
The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…
The Local Structure Theorem (LST) for Graph Minors roughly states that for every $H$-minor-free graph $G$ that contains a sufficiently large wall $W$, there is a small vertex subset $A,$ whose removal yields a graph that admits an "almost…
The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the…
The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial…
The Vapnik-\v{C}ervonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi, Estellon,…
Let ${H}=(V, {E})$ be a hypergraph on the vertex set $V$ and edge set ${E}\subseteq 2^V$. We show that number of distinct {\it traces} on any $k-$ subset of $V$, is most $k.{\hat \alpha}(H)$, where ${\hat \alpha}(H)$ is the {\it degeneracy}…
We show an $\Omega\big(\Delta^{\frac{1}{3}-\frac{\eta}{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(\Delta^{1+\eta}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In…
The local metric dimension ${\rm dim}_l$ in relation to the clique number $\omega$ is investigated. It is proved that if $\omega(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the…
A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in…
Let $G$ be a graph and $t\ge 0$. A new graph parameter termed the largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is introduced. Specifically, ${\hat\beta}_t(G)$ is the largest, overall $t$-shallow…
Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of…
Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ \omega(G) $. It has been conjectured that if $ n(G) \geq \omega(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{\omega(G) - 2}{\omega(G) -…
A box is the cartesian product of real intervals, which are either bounded or equal to $\mathbb{R}$. A box is said to be $d$-local if at most $d$ of the intervals are bounded. In this paper, we investigate the recently introduced local…
The min-diameter of a directed graph $G$ is a measure of the largest distance between nodes. It is equal to the maximum min-distance $d_{min}(u,v)$ across all pairs $u,v \in V(G)$, where $d_{min}(u,v) = \min(d(u,v), d(v,u))$. Our work…