Related papers: Spanning Properties of Theta-Theta Graphs
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively…
The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of…
In this paper, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with $k$ cut vertices (respectively, $k$ cut edges) with the minimal distance…
Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…
For any constants $d\ge 1$, $\epsilon >0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$,…
A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…
Token graphs, or symmetric powers of graphs, see \cite{alavi2002survey} and \cite{Fabila-Monroy2012}, are defined on the $k$-combinations of the vertex set of some graph $L$, where edges exist between two such combinations, if their…
We prove that every weighted graph contains a spanning tree subgraph of average stretch O((log n log log n)^2). Moreover, we show how to construct such a tree in time O(m log^2 n).
Prevailing methods for graphs require abundant label and edge information for learning. When data for a new task are scarce, meta-learning can learn from prior experiences and form much-needed inductive biases for fast adaption to new…
A $t$-spanner of a graph $G$ is a subgraph $H$ in which all distances are preserved up to a multiplicative $t$ factor. A classical result of Alth\"ofer et al. is that for every integer $k$ and every graph $G$, there is a $(2k-1)$-spanner of…
A spanner graph on a set of points in $R^d$ contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. In this paper we investigate new models and aim to interpret why good…
We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we…
A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…
A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$…
$t$-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a $t$-approximation on the distance of any two…
Treewidth is an important structural graph parameter that quantifies how closely a graph resembles a tree-like structure. It has applications in many algorithmic and combinatorial problems. In this paper, we study the treewidth of outer…
The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper…
In the spanning tree congestion problem, given a connected graph $G$, the objective is to compute a spanning tree $T$ in $G$ that minimizes its maximum edge congestion, where the congestion of an edge $e$ of $T$ is the number of edges in…
We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The…
In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least…