On Minimal and Minimum Hull Sets

International Journal of Mathematics and Mathematical Sciences, 2012

Given a collection of minimal graphs, M 1 , M 2 ,. .. , M n , with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions on M 1 , M 2 ,. .. , M n so that a convex combination of them will be a minimal graph. We will then provide two examples, taking a convex combination of Scherk's doubly periodic surface with the catenoid and Enneper's surface, respectively.

Discrete Mathematics, 1985

A set of points S of a graph is convex if any geodesic joining two points of S lies entirely within S. The convex hull of a set T of points is the smallest convex set that contains T. The hull number (h) of a graph is the cardinality of the smallest set of points whose convex hull is the entire graph. Characterisations are given for graphs with particular values of h, and upper and lower bounds for h are derived.

Computing Research Repository, 2011

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent sets. Recently, it was established that ker(G) is a subset of core(G) is true for every graph, while the corresponding equality holds for bipartite graphs. In this paper we present various structural properties of ker(G). The main finding claims that ker(G) is equal to the union of all inclusion minimal independent sets with positive difference.

Discrete Mathematics, 2005

The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighbourhoods are complete graphs. It is known that the class of graphs with the Minkowski-Krein-Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbour is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set.

Discrete Mathematics, 2015

A subset S of vertices of a graph G is g-convex if whenever u and v belong to S, all vertices on shortest paths between u and v also lie in S. The g-spectrum of a graph is the set of sizes of its g-convex sets. In this paper we consider two problems -counting g-convex sets in a graph, and determining when a graph has g-convex sets of every cardinality (such graphs are said to have the continuum property). We show that the problem of counting g-convex sets of a graph whose components have diameter at most 2 is #P-complete, but for the class of cographs these sets can be enumerated in linear time. The problem of determining whether or not the g-convexity of a graph has the continuum property is proven to be NP-complete. While every graph is shown to be an induced subgraph of a graph whose gconvexity possesses the continuum property, graphs with the continuum property are rare since for any fixed ϵ ∈ (0, 1) it is shown that almost all n-vertex graphs have a gap in their g-spectrum of size at least Ω(n 1-ϵ ). Moreover, it is shown that for almost all graphs, every g-convex set is a clique, from which it follows that the number of g-convex sets in a random graph is at least n c ln n for some constant c. The graph convexity under discussion fits within the class of alignments on a finite set, namely those set systems on a finite set V that contain the whole set, the empty set, and are closed under intersection. Finite topologies are perhaps the most famous examples of alignments, and our results here are compared and contrasted with what can be said for topologies on a finite set.

2020

We study R^2⊕R-separately convex hulls of finite sets of points in R^3, as introduced in <cit.>. When R^3 is considered as a certain subset of 3× 2 matrices, this notion of convexity corresponds to rank-one convex convexity K^rc. If R^3 is identified instead with a subset of 2× 3 matrices, it actually agrees with the quasiconvex hull, due to a recent result <cit.>. We introduce "2+1 complexes", which generalize T_n constructions. For a finite set K, a "2+1K-complex" is a 2+1 complex whose extremal points belong to K. The "2+1-complex convex hull of K", K^cc, is the union of all 2+1K-complexes. We prove that K^cc is contained in the 2+1 convex hull K^rc. We also consider outer approximations to 2+1 convexity based in the locality theorem <cit.>. Starting with a crude outer approximation we iteratively chop off "D-prisms". For the examples in <cit.>, and many others, this procedure reaches a "2+1K-complex" in a ...

Electronic Notes in Discrete Mathematics, 2013

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u, v ∈ V of a connected undirected graph G = (V, E), the closed interval I[u, v] of u and v is the set of vertices that belong to some shortest (u, v)-path. For any S ⊆ V , let I[S] = u,v∈S I[u, v]. A subset S ⊆ V is (geodesically) convex if I[S] = S. Given a subset S ⊆ V , the convex hull I h [S] of S is the smallest convex set that contains S. We say that S is a hull set of G if I h [S] = V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G). First, we show a polynomial-time algorithm to compute the hull number of any P 5-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

One classical result of Freimann gives the optimal lower bound for the cardinality of A + A if A is a d-dimensional finite set in R d . Matolcsi and Ruzsa have recently generalized this lower bound to |A + kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.

International Journal of Mathematical Analysis, 2014

Given a connected graph G, a subset C of V (G) is called a weakly convex set of G if for every two vertices u, v ∈ C, there exists a u-v geodesic whose vertices belong to C or equivalently, if for every two vertices u, v ∈ C, d C (u, v) = d G (u, v). Let S be a subset of V (G). A weakly convex hull of S is a weakly convex set of minimum order containing S. In this paper, we introduce the concepts of weakly convex hull and weakly hull number of a graph. Moreover, we determine the weakly hull numbers of some special graphs and graphs resulting from some binary operations.

Journal of Combinatorial Theory, Series A, 2011

Let k, d, λ 1 be integers with d λ. What is the maximum positive integer n such that every set of n points in R d has the property that the convex hulls of all k-sets have a transversal (d − λ)-plane? What is the minimum positive integer n such that every set of n points in general position in R d has the property that the convex hulls of all k-sets do not have a transversal (d − λ)plane? In this paper, we investigate these two questions. We define a special Kneser hypergraph and, by using some topological results and the well-known λ-Helly property, we relate our second question to the chromatic number of such hypergraphs. Moreover, we establish a connection (when λ = 1) with Kneser's conjecture, first proved by Lovász. Finally, we prove a discrete flat center theorem.