Related papers: On Explicit Constructions of Extremely Depth Robus…
A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small…
Given an increasing graph property $\mathcal{P}$, a graph $G$ is $\alpha$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-\alpha)$-proportion of its neighbours,…
Graph embedding provides an efficient solution for graph analysis by converting the graph into a low-dimensional space which preserves the structure information. In contrast to the graph structure data, the i.i.d. node embedding can be…
Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…
A temporal graph $\mathcal{G}=(G,\lambda)$ can be represented by an underlying graph $G=(V,E)$ together with a function $\lambda$ that assigns to each edge $e\in E$ the set of time steps during which $e$ is present. The reachability graph…
A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for…
Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined…
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…
We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…
Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have…
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the…
Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly…
In this work, we study the equivalence of various robustness properties of toric ideals of weighted oriented graphs. For any weighted oriented graph $D$, if its toric ideal $I_D$ is generalized robust (or weakly robust), then we show that…
A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$…
Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H…
A graph $G$ is $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a total…
This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…
A graph $G=(V,E)$ is a geometric intersection graph if every node $v \in V$ is identified with a geometric object of some particular type, and two nodes are adjacent if the corresponding objects intersect. Geometric intersection graph…
Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured…
A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that…