A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the cardinality of a minimum geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also stablish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.
Let $D$ be an orientation of a simple graph. Given $u,v\in V(D)$, a directed shortest $(u,v)$-path is a $(u,v)$-geodesic. $S \subseteq V(D)$ is convex if, for every $u,v \in S$, the vertices in each $(u,v)$-geodesic and in each $(v,u)$-geodesic are in $S$. For each $S \subseteq V(D)$ the (convex) hull of $S$, denoted by $[S]$, is the smallest convex set containing $S$. $S \subseteq V(D)$ is a hull set if $[S] = V(D)$. $S \subseteq V(D)$ is a geodetic set of $D$ if each vertex of $D$ lies in a $(u,v)$-geodesic, for some $u,v \in S$. The cardinality of a minimum hull set (resp. geodetic set) of …
We prove three results conjectured or stated by Chartrand, Fink and Zhang [European J. Combin {\bf 21} (2000) 181--189, Disc. Appl. Math. {\bf 116} (2002) 115--126, and pre-print of ``The hull number of an oriented graph'']. For a digraph $D$, Chartrand et al. defined the geodetic, hull and convexity number -- $g(D)$, $h(D)$ and $con(D)$, respectively. For an undirected graph $G$, $g^{-}(G)$ and $g^{+}(G)$ are the minimum and maximum geodetic numbers over all orientations of $G$, and similarly for $h^{-}(G)$, $h^{+}(G)$, $con^{-}(G)$ and $con^{+}(G)$. Chartrand and Zhang gave a proof that $g^{…
Let $G$ be a finite, simple, and undirected graph and let $S$ be a set of vertices of $G$. In the geodetic convexity, a set of vertices $S$ of a graph $G$ is convex if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The convex hull $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a hull set. The cardinality $h(G)$ of a minimum hull set of $G$ is the hull number of $G$. The complementary prism $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect…
College of Saint Benedict and Saint John’s University（圣本笃学院和圣约翰大学）；Northern Arizona University（北亚利桑那大学）；University of South Florida（南佛罗里达大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）
The {\em Kneser graph} $K(2n+k,n)$, for positive integers $n$ and $k$, is the graph $G=(V,E)$ such that $V=\{S\subseteq\{1,\ldots,2n+k\} : |S|=n\}$ and there is an edge $uv\in E$ whenever $u\cap v=\emptyset$. Kneser graphs have a nice combinatorial structure, and many parameters have been determined for them, such as the diameter, the chromatic number, the independence number, and, recently, the hull number (in the context of $P_3$-convexity). However, the determination of geodetic convexity parameters in Kneser graphs still remained open. In this work, we investigate both the geodetic number …
The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and…
The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that ${\rm sg}(G \,\square\, K_2) \geq {\rm sg}(G)$ holds for different families of graphs,…
A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If $G$ is a graph, then ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that one can assign a fixed geodesic to each pair $\{x,y\}\subseteq S$ so that these $\binom{|S|}{2}$ geodesics cover all the vertices of $G$. In this paper, the strong geodesic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on flat grids and flat cylinders.
Let $G=(V(G),E(G))$ be a simple graph. A set $S \subseteq V(G)$ is a strong edge geodetic set if there exists an assignment of exactly one shortest path between each pair of vertices from $S$, such that these shortest paths cover all the edges $E(G)$. The cardinality of a smallest strong edge geodetic set is the strong edge geodetic number $\text{sge}(G)$ of $G$. In this paper, the strong edge geodetic problem is studied on the Cartesian product of two paths. The exact value of the strong edge geodetic number is computed for $P_n \, \square \, P_2$, $P_n \, \square \, P_3$ and $P_n \, \square …
A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of $G$ such that every vertex of $G$ belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order o…
College of Saint Benedict and Saint John’s University（圣本笃学院与圣约翰大学）；Northern Arizona University（北亚利桑那大学）；Lewis University（路易斯大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）
We present formulas to compute the P3-geodetic number, the P3-hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit interval graph and a parameter involving the diameter of a unit interval graph related to it. Finally, we present a hereditary graph class, defined by forbidden induced subgraphs, such that its percolation time is equal to one.
We prove that a connected bipartite graph G is a partial cube if and only if the set of attaching points of any copoint of G is convex. A consequence of this result is that any connected bipartite graph with pre-hull number at most 1 is a partial cube. We show that the class of partial cubes with pre-hull number at most 1 is closed under gated subgraphs, gated amalgams and cartesian products.
The Closed Geodetic Game: algorithms and strategies
Université Clermont Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Mines de Saint-Étienne（圣艾蒂安国立高等矿业学校）；Clermont-Auvergne-INP（克莱蒙奥弗涅理工学院）；LIMOS（极限与优化实验室）；INRAE（法国国家农业、食品和环境研究所）；Laboratoire G-SCOP（G-SCOP实验室）；CNRS, Univ. Grenoble Alpes（法国国家科学研究中心，格勒诺布尔阿尔卑斯大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）；Indian Institute of Technology Dharwad（印度理工学院达瓦德分校）；Polytechnic University (Kyaing Tong)（桂开工科大学）
Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest $P_3$-convex set of $G$ that contains $S$. Carathéodory number of a graph $G$, denoted by $c(G)$, is the smallest integer $c$ such that for every subset $S \subseteq V(G)$ and every vertex $p$ in the $P_3$-hull of $S$, there exists a subset $F \subseteq S$ with $|F| \leq c$ such that $p$ belongs to the $P_3$-hull of $F$. In this article, we present upper bounds and fo…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes.
Peano partial cubes are the bipartite graphs whose geodesic interval spaces are (closed) join spaces. They are the partial cubes all of whose finite convex subgraphs have a pre-hull number which is at most 1. Special Peano partial cubes are median graphs, cellular bipartite graphs and netlike partial cubes. Analogous properties of these graphs are satisfied by Peano partial cubes. In particular the convex hull of any isometric cycle of such a graph is a gated quasi-hypertori (i.e., the Cartesian product of copies of K_2 and even cycles). Moreover, for any Peano partial cubes G that contains no…
We study the convex hull of the graph of a quadratic function $f(\mathbf{x})=\sum_{ij\in E}x_ix_j$, where the sum is over the edge set of a graph $G$ with vertex set $\{1,\dots,n\}$. Using an approach proposed by Gupte et al. (Discrete Optimization $\textbf{36}$, 2020, 100569), we investigate minimal extended formulations using additional variables $y_{ij}$, $1\leq i<j\leq n$, representing the products $x_ix_j$. The basic idea is to identify a set of facets of the Boolean Quadric Polytope which is sufficient for characterizing the convex hull for the given graph. Our main results are extended …
We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $R^d$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: 1) Find t…
Consider a sequence of partial sums $S_i= ξ_1+\dots+ξ_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $ξ_1,\dots,ξ_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(ξ_1,\dots,ξ_n)$ is exchangeable and a certain general position condition holds, we prove that the expected number of $k$-dimensional faces of $C_n$ is given by the formula $$ \mathbb E [f_k(C_n)] = \frac{2\cdot k!}{n!} \sum_{l=0}^{\infty}\genfrac{[}{]}{0pt}{}{n+1}{d-2l} \genfrac{\{}{\}}{0pt}{}{d-2l}{…
Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between $u$ and $v$ in $G$ is a walk of the form $T: u,w_1,\ldots,w_k,v,$ where $k\ge 1$, in which $w_1$ is the only neighbor of $u$ in $T$ and $w_k$ is the only neighbor of $v$ in $T$. As in geodesic or monophonic convexity, toll interval between $u,v\in V(G)$ is a set $T_G(u,v)=\{x\in V(G)\,:\,x \textrm{ lies on a tolled walk between } u \textrm{ and } v\}$. A set of vertices $S$ is toll convex, if $T_{G}(u,v)\subseteq S$ for all $u,v\in S$. First part of the paper reinvestigates the characterization of conv…
A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\in V(G)$ is a set $T_G(u,v)=\{x\in V(G)~|~x \textrm{ lies on a tolled walk between } u \textrm{\, and\,} v\}$. A set $S \subseteq V(G)$ is toll convex, if $T_{G}(u,v)\subseteq S$ for all $u,v\in S$. A toll closure of a set $S \subseteq V(G)$ is the union of toll intervals between all pairs of vertices from $S$. The size of a smallest set $S$ whose toll closur…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph c…
