Related papers: Representing Graphs via Pattern Avoiding Words
Isomorphisms allow human cognition to transcribe a potentially unsolvable problem from one domain to a different domain where the problem might be more easily addressed. Current approaches only focus on transcribing structural information…
A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type…
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For…
Continuous word representations, trained on large unlabeled corpora are useful for many natural language processing tasks. Popular models that learn such representations ignore the morphology of words, by assigning a distinct vector to each…
In [1], whether a target binary string s can be represented from a boolean formula with operands chosen from a set of binary strings W was studied. In this paper, we first examine selecting a maximum subset X from W, so that for any string…
A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that…
An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total…
Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a \v{S}olt\'es vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, \v{S}olt\'es posed the…
Graph is considered neutral if its assortativity coefficient $r$ is equal to zero. In this paper, we address an outstanding conjecture, i.e., whether is there a neutral graph on $n$ vertices? First, we show that for $n\geq7$, there is at…
Graph are a ubiquitous data representation, as they represent a flexible and compact representation. For instance, the 3D structure of RNA can be efficiently represented as $\textit{2.5D graphs}$, graphs whose nodes are nucleotides and…
A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in…
Abusive behaviors are common on online social networks. The increasing frequency of antisocial behaviors forces the hosts of online platforms to find new solutions to address this problem. Automating the moderation process has thus received…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained…
Learning knowledge graph embedding from an existing knowledge graph is very important to knowledge graph completion. For a fact $(h,r,t)$ with the head entity $h$ having a relation $r$ with the tail entity $t$, the current approaches aim to…
Representation learning on text-attributed graphs (TAGs) has attracted significant interest due to its wide-ranging real-world applications, particularly through Graph Neural Networks (GNNs). Traditional GNN methods focus on encoding the…
We show that if the Sch\"{u}tzenberger graph of every positive word, that contains an $R$-word only once as it's subword, is finite over an Adain presentation $\langle X|u=v\rangle$, then the Sch\"{u}tzenberger graph of every positive word…
String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be done automatically by double-pushout rewriting. !-graphs…
A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight.…