Given a graph $G=(V,E)$, the maximum bond problem searches for a maximum cut $δ(S) \subseteq E$ with $S \subseteq V$ such that $G[S]$ and $G[V\setminus S]$ are connected. This problem is closely related to the well-known maximum cut problem and known under a variety of names such as largest bond, maximum minimal cut and maximum connected (sides) cut. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While cut polytopes have been intensively studied, there are no results on bond polytopes. We start a structural study of the latter.
We investigate the relation between cut- and bond polytopes and study the effect of graph modifications on bond polytopes and their facets. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and $3$-connected planar $(K_5-e)$-minor free graphs. Moreover we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on $3$-connected graphs. This yields a linear time algorithm for maximum bond on $(K_5-e)$-minor free graphs.
In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope $X$ on $k$ vertices such that $A\subseteq X \subseteq B$. The polytope $A$ is given by a set of vertices and the polytope $B$ is given by the defining hyperplanes. We show a universality theorem for NPP. Given an instance $I$ of the NPP, we define the solutions set of $I$ as $$ V'(I) = \{(x_1,\ldots,x_k)\in \mathbb{R}^{k\cdot n} : A\subseteq \text{conv}(x_1,\…
Given a graph $G=(V,E)$ and a set $S \subseteq \binom{V}{2}$ of terminal pairs, the minimum multicut problem asks for a minimum edge set $δ\subseteq E$ such that there is no $s$-$t$-path in $G -δ$ for any $\{s,t\}\in S$. For $|S|=1$ this is the well known $s$-$t$-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope $\text{MultC}^\square (G,S)$ is the convex hull of all multicuts in $G$; the multicut dominant is given by $\text{MultC}(G,S)=\text{MultC}^\square (G,S)+\mathbb{R}^E$. The latter is the relevant object for …
A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $γ(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set edges $A\subseteq E(G)$ such that $γ(G-A)=γ(G)+1$. A chordal graph is a graph with no induced cycle of length four or more. In this paper, we prove that the bondage number of a chordal graph $G$ is at most the order of its maximum clique, that is, $b(G)\leq ω(G)$. We show that this bound is best possible.
A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $γ(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set of edges $A\subseteq E(G)$, such that $γ(G-A)=γ(G)+1$. The $d$-Bondage is the problem of deciding, given a graph $G$ and an integer $d\geq 1$, if $b(G)\leq d$. This problem is known to be $\mathsf{NP}$-hard even for bipartite graphs and $d=1$. In this paper, we show that $1$-Bondage is $\mathsf{NP}$-hard, even for the class of $3$-regular …
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diame…
We study polytopes defined by inequalities of the form $\sum_{i\in I} z_{i}\leq 1$ for $I\subseteq [d]$ and nonnegative $z_i$ where the inequalities can be reordered into a matrix inequality involving a column-convex $\{0,1\}$-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs $G$ with a Hamiltonian path, which we call spinal graphs. We show that the volume of these flow polytopes is the number of extensions …
Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A set $\tilde{D} \subseteq V$ of a graph $G=(V,E) $ is called an outer-connected dominating set for $G$ if (1) $\tilde{D}$ is a dominating set for $G$, and (2) $G [V \setminus \tilde{D}]$, the induced subgraph of $G$ by $V \setminus \tilde{D}$, is connected. The minimum size among all outer-connected dominating sets of $G$ is called the outer-connected domination number of $G$ and is denoted by $\tildeγ_c(G)$. We define the outer-connected bondage number of a…
In a graph, a Clique-Stable Set separator (CS-separator) is a family $\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a cut $( W,W')$ in $\mathcal{C}$ such that $K \subseteq W$ and $S \subseteq W'$. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [17] asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive …
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted by $γ(G)$, is the smallest cardinality of a dominating set of $G$. The bondage number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with domination number larger than $γ(G)$. The reinforcement number of $G$ is the smallest number of edges whose addition to $G$ results in a graph with smaller domination number than $γ(G)$. In 2012, Hu and Xu proved that the decision problems for …
A polytope $P$ is a {\em model} for a combinatorial problem on finite graphs $G$ whose variables are indexed by the edge set $E$ of $G$ if the points of $P$ with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) MaxCut Problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of $G$. In this paper we introduce a new polytope $\mathbb{P}_{12} \subset \mathbb{R}^{|E|}$ given by at most $11|E|$ inequalities, which is a …
A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq {\cal F}$ of sets is a copy of a poset $P$ in ${\cal F}$ if there exists a bijection $ϕ:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that whenever $x \le_P x'$ holds, then so does $ϕ(x)\subseteq ϕ(x')$. For a family ${\cal F}$ of sets, let $c(P,{\cal F})$ denote the number of copies of $P$ in ${\cal F}$, and we say that ${\cal F}$ is $P$-free if $c(P,{\cal F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families ${\cal F} \subseteq 2^{[n]}$, i.e. $\max\{c(Q,{\cal F}): {\cal…
A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of $s$ linear equalities and inequalities in $d_1 + d_2$ variables for some $s$. We show that for a fixed $k$, the number of edges in a $K_{k,k}$-free semilinear $H$ is almost linear in $n$, namely $|E| = O_{s,k,\varepsilon}(n^{1+\varepsilon})$ for any $\varepsilon > 0$; and more generally, $|E| = O_{s,k,r,\varepsilon}(n^{r-1 + \…
