Related papers: Existence of most reliable two-terminal graphs wit…
A two-terminal graph is a graph G equipped with two vertices in V(G) called terminals. Let T(n,m) be the set of two-terminal graphs on n vertices and m edges. Let G be in T(n,m) and let p be in [0,1]. The two-terminal reliability of G at p,…
A two-terminal graph is a graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the set of all nonisomorphic connected simple two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph…
A graph $G$ with $k$ specified target vertices in vertex set is a $k$-terminal graph. The $k$-terminal reliability is the connection probability of the fixed $k$ target vertices in a $k$-terminal graph when every edge of this graph survives…
If $G$ is a simple graph and $\rho\in[0,1]$, the reliability $R_G(\rho)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $\rho$. A simple graph $G$ is a \emph{uniformly most…
The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices…
If $G$ is a simple graph and $\rho\in[0,1]$, the reliability $R_G(\rho)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $\rho$. A simple graph $G$ is a \emph{uniformly most…
A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability $p$. One can ask for the probability that all vertices can communicate ({\em all-terminal…
Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ are operational independently with probability $p \in[0,1]$. For fixed vertices $s$ and $t$, the \emph{two-terminal reliability} of $G$ is the probability…
A key issue in network reliability analysis. A graph with $n$ nodes and whose $e$ edges fail independently with probability $p$ is an \emph{Uniformly Most Reliable Graph} (UMRG) if it has the highest reliability among all graphs with the…
A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is…
Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$…
For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected…
A graph $G = (V,E)$ is globally rigid in $\mathbb{R}^d$ if for any generic placement $p : V \rightarrow \mathbb{R}^d$ of the vertices, the edge lengths $||p(u) - p(v)||, uv \in E$ uniquely determine $p$, up to congruence. In this paper we…
A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…
Let $r(u,v)$ be the resistance distance between two vertices $u, v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$…
An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…
Given a graph $G$ and a set of terminals $T$, a \emph{distance emulator} of $G$ is another graph $H$ (not necessarily a subgraph of $G$) containing $T$, such that all the pairwise distances in $G$ between vertices of $T$ are preserved in…
Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any…
Let $G=(V,E)$ be a simple graph with $|V|=n$ nodes and $|E|=m$ links, a subset $K \subseteq V$ of \emph{terminals}, a vector $p=(p_1,\ldots,p_m) \in [0,1]^m$ and a positive integer $d$, called \emph{diameter}. We assume nodes are perfect…
A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…