Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, and mu(G) is the size of a maximum matching. The number id_{c}(G)=max{|I|-|N(I)|:I is an independent set} is called the critical independence difference of G, and A is critical if |A|-|N(A)|=id_{c}(G). We define core(G) as the intersection of all maximum independent sets, and ker(G)as the intersection of all critical independent sets. In this paper we prove that if a graph G is non-quasi-regularizable (i.e., there exists some independent set A, such that |A|>|N(A)|), then: ker(G) is a subset of core(G), and |ker(G)|> id_{c}(G) >= alpha(G)-mu(G) > 0.
We characterize the vertices belonging to all minimum dominating sets, to some minimum dominating sets but not all, and to no minimum dominating set. We refine this characterization for some well studied sub-classes of graphs: chordal, claw-free, triangle-free. Also we exhibit some graphs answering to some open questions of the literature on minimum dominating sets.
Given a non empty set $S$ of vertices of a graph, the partiality of a vertex with respect to $S$ is the difference between maximum and minimum of the distances of the vertex to the vertices of $S$. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of …
For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\displaystyle\max_{x \in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center …
We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number $n$ of maximal independent sets for a class of natural numbers that can be represented as concatenation of periodic binary words.
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…
For a given undirected graph $G$, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in $S$. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let $Q^n$ denote the $n$-dimensional hypercube with vertex set ${0,1}^n$. In Erdös and Renyi (Erdos & Renyi…
Every triangle-free planar graph on n vertices has an independent set of size at least (n+1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides whether G has an independent set of size at least (n+k)/3, in time 2^{O(sqrt{k})}n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists epsilon>0 such that every planar graph of girth at least five on n vertices has an independent…
The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for $P_6$-free graphs is unknown. In this note, we show that the MWIS problem can be solved in time $O(n^3m)$ for ($P_6$, banner)-free graphs by analyzing the structure of subclasses of these class of graphs. This extends the existing results for ($P_5$, banner)-free graphs, and ($P_6$, $C_4$)-free graphs. Here, $P_t$ denotes the chordless path on $t$ vertices, and a banner is the graph obtained from a chordl…
In this paper we study the the average order of dominating sets in a graph, $\operatorname{avd}(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs $G$ of order $n$ without isolated vertices, $\operatorname{avd}(G) \leq 2n/3$. Recently, Erey (2021) proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $\operatorname{avd}(G) = 2n/3$. We also use our bounds to prove the average version of Vizing's Conjecture.
Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least \[ e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)} \] independent sets. This improves a recent result of the first and third authors \cite{countingind}. In particular, it implies that as $n \to \infty$, every triangle-free graph on $n$ vertices has at least $e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138..$. Further, we show that for all $n$, there exists a triangle-free graph with $n$ vertices which has at most $e^{(c_2+o(1))\…
A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $γ^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin…
Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1's in the binary representatio…
Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ a…
The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.
This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing sets of $G$ with an edge between vertices $B$ and $B'$ of $\mathscr{Z}(G)$ if and only if $B$ can be obtained from $B'$ by changing a single vertex of $G$. It is shown that the zero forcing graph of a forest is connected, but that many zero forcing graphs are disconnected. We characterize the base graphs whose zero forcing graphs are either a path or the comp…
A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in $O(1.7159^n)$ time. It is known that the number of minimal dominating sets for interval graphs and trees on $n$ vertices is at most $3^{n/3} \approx 1.4422^n$. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set $D$, denoted by $\mathcal{C}_D…
Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a minimum set of vertices such that after iteratively applying the rule, all of the vertices become colored (i.e., a minimum zero forcing set). Of particular interest is the propagation time of a chosen set which is the number of steps the rule must be applied in order to color all the vertices of a graph. We give a purely linear algebraic interpretation of zer…
A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. Following analogous studies in the literature related to independent sets, dominating sets, and total dominating sets, we study in this paper the class of graphs in which the connected dominating sets can be separated from the other vertex subsets by a linear weight function. More precisely, we say that a graph is connected-domishold if it admits non-negative real weights associated to its vertices such that a set of vertices is a connected dominating set if and only if the sum of the corre…
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…
A non-empty set $X$ of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by $X$ is connected and it is called convex if no two vertices of $X$ are connected by a directed path in which some vertices are not in $X$. The set of convex sets (connected convex sets) of an acyclic digraph $D$ is denoted by $\sco(D)$ ($\scc(D)$) and its size by $\co(D)$ ($\cc(D)$). Gutin, Johnstone, Reddington, Scott, Soleimanfallah, and Yeo (Proc. ACiD'07) conjectured that the sum of the sizes of all (connected) convex sets in $D$ equals $Θ(n \cdot \co(D))$ ($Θ(n \cdot \cc…
Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a shortest directed $u-v$ path in $D$. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph $D$ are defined with respect to this metric. The main aim of this paper is to identify the above said metrically defined sets of a large strong digraph $D$ in terms of its prime factor decomposition with respect to cartesian pro…
A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the study of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm fo…
This work introduces a novel algorithm for finding the connected components of a graph where the vertices and edges are grouped into sets defining a Set--Based Graph. The algorithm, under certain restrictions on those sets, has the remarkable property of achieving constant computational costs with the number of vertices and edges. The mentioned restrictions are related to the possibility of representing the sets of vertices by intension and the sets of edges using some particular type of maps. While these restrictions can result strong in a general context, they are usually satisfied in the pr…
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum number of vertices that can be left fixed in the worst case. In the \emph{allocation problem}, we are given a planar graph $G$ on $n$ vertices together with an $n$-point set $X$ in the plane and have to draw $G$ without edge crossings so that as many vertices as possible are located in $X$. Let $fit(G)$ denote the maximum number of points fitting this purpos…
