The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position problem called the \emph{monophonic position problem}, obtained by replacing `shortest path' by `induced path'. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifyin…
We present some complexity results concerning the problems of covering a graph with $p$ convex sets and of partitioning a graph into $p$ convex sets. The following convexities are considered: digital convexity, monophonic convexity, $P_3$-convexity, and $P_3^*$-convexity.
We consider a natural combinatorial optimization problem on chordal graphs, the class of graphs with no induced cycle of length four or more. A subset of vertices of a chordal graph is (monophonically) convex if it contains the vertices of all chordless paths between any two vertices of the set. The problem is to find a maximum-weight convex subset of a given vertex-weighted chordal graph. It generalizes previously studied special cases in trees and split graphs. It also happens to be closely related to the closure problem in partially ordered sets and directed graphs. We give the first polyno…
A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.
The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r…
Erdős asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? Füredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, Tóth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in genera…
The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural charac…
General position sets in strong products with paths and cycles
Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那数学、物理与力学研究所）
Given a set of $n$ points in $R^2$, the Szemerédi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position---no two parallel, no three meeting at a point---then one can get a much tighter bound. We prove: for every $ε> 0$ there exists some $δ> 0$ such that for sufficiently large values of $n$, every set of lines in general position, each intersecting an $n \times n$ grid of points in at least $n^{…
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the co…
For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction, based on iterated Minkovski sums, is used to obtain new lower bounds for $S_{P}(n,m)$ when $P$ is an arbitrary pattern. Improved bounds are obtained when $P$ is a triangle or a regular polygon with few sides. It is also shown that $S_{P}(n,m)\geq n^{2-ε}$ whenever $m(n)\to \infty$ as $n \to\infty$. Finite sets with no collinear triples and not containing the 4 …
A flag of a finite set $S$ is a set $f$ of non-empty proper subsets of $S$ such that $A\subseteq B$ or $B\subseteq A$ for all $A,B\in f$. The set $\{|A|:A\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are in general position (with respect to $S$) when $A\cap B=\emptyset$ or $A\cup B=S$ for all $A\in f$ and $B\in f'$. We study sets of flags of a fixed type $T$ that are mutually not in general position and are interested in the largest cardinality of these sets. This is a generalization of the classical Erdös-Ko-Rado problem. We will give some basic facts and determine the largest ca…
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P \cup R$ is contained in a cubic curve $c$ in the plane, then $P$ has a special property with respect to the natural group action on $c$. That is, $P$ is contained in a coset of a subgroup $H$ of $c$ of cardinality at most $|R|$. We use the same approach to show a similar result in the case where each of $B$ and $G$ is a set of $n$ points in general position …
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum ver…
In 1935, Erdős and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a set of $n:=2^{t-2}$ points in general position in the plane, such that every convex polygon with vertices in this set has at most $\log_2(n)+1$ vertices. In this paper we show how to realize their construction in an integer grid of size $O(n^2 \log_2(n)^3)$.
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…
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We show that $P \cup R$ must be contained in a cubic curve.
A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. O…
A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be the set of general position sets of a graph $G$. The general position number of a graph $G$, denoted by $gp(G)$, is defined as $gp(G)=\max\{|R|:R\in\mathcal{R}\}$. In this paper, we determine the bounds on the gp-numbers for any maximal outerplane graph and characterize the corresponding extremal graphs.
We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order logic formulae concerning some sets of multisets of uniform cardinality over boolean variables would prove the Erdös-Szekeres conjecture, which states that for any set of 2^(n-2)+1 points in general position, there exists n points forming a convex polygon, where n is greater than or equal to 3.
Let $B$ and $R$ be point sets (of {\em blue} and {\em red} points, respectively) in the plane, such that $P:=B\cup R$ is in general position, and $|P|$ is even. A line $\ell$ is {\em balanced} if it spans one blue and one red point, and on each open halfplane of $\ell$, the number of blue points minus the number of red points is the same. We prove that $P$ has at least $\min \{|B|,|R|\} $ balanced lines. This refines a result by Pach and Pinchasi, who proved this for the case $|B|=|R|$.
We introduce a new type of positional games, played on a vertex set of a graph. Given a graph $G$, two players claim vertices of $G$, where the outcome of the game is determined by the subgraphs of $G$ induced by the vertices claimed by each player (or by one of them). We study classical positional games such as Maker-Breaker, Avoider-Enforcer, Waiter-Client and Client-Waiter games, where the board of the game is the vertex set of the binomial random graph $G\sim G(n,p)$. Under these settings, we consider those games where the target sets are the vertex sets of all graphs containing a copy of …
Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the numbers $X_{k,\ell}(S)$. This problem is related to the Erdős-Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.
For a finite set $A\subset \mathbb{R}^d$, let $Δ(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $γ_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $\mathbb{R}^d$, satisfying $Δ(A) \leq αn^{1/d}$ for a fixed $α>0$, contains at least $γ_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $γ_2(n)=Θ(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensi…
A Skolem sequence is a sequence a_1,a_2,...,a_2n (where a_i \in A = {1,...,n }), each a_i occurs exactly twice in the sequence and the two occurrences are exactly a_i positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The problem of deciding which sets of the form A = {1,...,n} are Skolem sets was solved by Thoralf Skolem in the late 1950's. We study the natural generalization where A is allowed to be any set of n positive integers. We give necessary conditions for the existence of Skolem sets of this generalized form. We conjecture these necessary …
In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bi…
We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.
Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each of them containing points of both colors. We also show that if $P$ is a set of $n(d+1)$ points in general position in $\mathbb{R}^d$ colored by $d$ colors with at least $n$ points of each color, then there exist $n$ pairwise disjoint $d$-dimensional simplices with vertices in $P$, each of them containing a point of every color. These results can be viewed a…
Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games, similar to those introduced by Holshouser, Reiter, Smith, Stănică, can be defined as having its sets of winning positions identical to well-known sets avoiding arithmetic progressions such as $x+z=2y$, studied by Szerkeres, Erd\H os and Turán, and many others, thus exploring a new territory combining ideas from combinatorial games and combinatorial number theory. …
We consider point sets in the real projective plane $\mathbb{R}P^2$ and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős--Szekeres theorem about point sets in convex position in $\mathbb{R}P^2$, which was initiated by Harborth and Möller in 1994. The notion of convex position in $\mathbb{R}P^2$ agrees with the definition of convex sets introduced by Steinitz in 1913. For $k \geq 3$, an (\affine) $k$-hole in a finite set $S \subseteq \math…
A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension $2$ are planar and therefore have at most $3n-6$ edges. We prove that all such graphs have at most $2n-3$ edges while on the other hand we present a class of non-planar graphs of strong convex dimension $2$. We also give lower bounds on the maximum number of edges a graph of strong convex dimension $2$ can have and discuss v…
We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.
