Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor\frac{n}{3}\rfloor \leq |X_i|\leq \lceil\frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1, 1\leq i\leq 3\}$. It is easy to check that the trace $\mathcal{G}_{\mid Y}:=\{G\cap Y\colon G\in \mathcal{G}\}$ satisfies $|\mathcal{G}_{\mid Y}|\leq 12$ for all 4-sets $Y\subset [n]$. For $n\geq 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $|\mathcal{F}|>|\mathcal{G}|$ then $|\mathcal{F}_{\mid C}|\geq 13$ for some $C\subset [n]$, $|C|=4$. Several further results of a similar flavor are established as well.
Let $γ(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $γ(G-v)=γ(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this paper, we correspondingly introduce two such definitions: (i) a set $S\subseteq V(G)$ is called a \emph{strong critical vertex-set} of $G$ if $γ(G-S)=γ(G)-|S|$; (ii) a graph $G$ is called \emph{strong $l$-vertex-sets-critical} if $V(G)$ can be partitioned into $l$ strong critical vertex-sets of $G$. Whereafter, we give some properties of strong $l$-vertex-sets-critical graph…
We introduce the concept of the primitivity of independent set in vertex-transitive graphs, and investigate the relationship between the primitivity and the structure of maximum independent sets in direct products of vertex-transitive graphs. As a consequence of our main results, we positively solve an open problem related to the structure of independent sets in powers of vertex-transitive graphs.
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented.
The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection digraph from a given digraph.
A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, \ell}(k)$ that meets all such minor-mod…
A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We characterize uniquely minimal sets of X: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of X: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate …
Crapo introduced a construction of interval partitions of the Boolean lattice for sets equipped with matroid structure. This construction, in the context of graphic matroids, is related to the notion of edge activities introduced by Tutte. This implies that each spanning subgraph of a connected graph can be constructed from edges of exactly one spanning tree by deleting a unique subset of internally active edges and adding a unique subset of externally active edges. Since the family of vertex independent sets does not give rise to a matroid structure we can not apply Crapo's construction on th…
We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis$_\bigtriangleup(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis$_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis$_{\bigtr…
The direct product $G\times H$ of graphs $G$ and $H$ is defined by: \[V(G\times H)=V(G)\times V(H)\] and \[E(G\times H)=\left\{[(u_1,v_1),(u_2,v_2)]: (u_1,u_2)\in E(G) \mbox{\ and\ } (v_1,v_2)\in E(H)\right\}.\] In this paper, we will prove that the equality $$α(G\times H)=\max\{α(G)|H|, α(H)|G|\}$$ holds for all vertex-transitive graphs $G$ and $H$, which provides an affirmative answer to a problem posed by Tardif (Discrete Math. 185 (1998) 193-200). Furthermore, the structure of all maximum independent sets of $G\times H$ are determined.
In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $Γ$ and a canonical subgeometry $Σ$ in $\mathrm{PG}(n-1,q^n)$, with $Γ\capΣ=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $Σ$ from the vertex $Γ$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the project…
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 …
A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if $G$ is a connected planar graph of girth at least five on $n$ vertices and $m$ edges, then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. By Euler's formula, this implies that $G$ has a feedback vertex set of size at most $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one o…
Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.
We consider the following generalization of the seminal Erdős-Ko-Rado theorem, due to Frankl. For k>= 2, let F be a k-wise intersecting family of r-subsets of an n element set X, i.e. any k sets in F have a nonempty intersection. If r<= (k-1/k)n, then |F|<={n-1 \choose r-1}. We extend Frankl's theorem in a graph-theoretic direction. For a graph G, and r>=1, let P^r(G) be the family of all r-subsets of the vertex set of G such that every r-subset is either an independent set or contains a maximum independent set. We will consider k-wise intersecting subfamilies of this family for the graph M_n,…
A parallel $d$-stable trace is a closed walk which traverses every edge of a graph exactly twice in the same direction and for every vertex $v$, there is no subset $X \subseteq N(v)$ with $1 \leq |N| \leq d$ such that every time the walk enters $v$ from $X$, it also exits to a vertex in $X$. In the past, $d$-stable traces were investigated as a mathematical model for an innovative biotechnological procedure -- self-assembling of polypeptide structures. Among other, it was proven that graphs that admit parallel $d$-stable traces are precisely Eulerian graphs with minimum degree strictly larger …
In this paper, we study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at $1$. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, while for trees we prove that the path minimises the average size of independent sets and the star maximises it. While removing a vertex does not always decrease the average size of independent sets, we prove that there always exists a vertex for whic…
The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$ to be colored (black) after some times. "color change rule": a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs w…
We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets, and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumera…
A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa-Karp-Rosenberg conjecture is that every monotone graph property is elusive for finite vertex sets. We show that the situation is completely different for infinite vertex sets: the monotone graph properties "every vertex has degree at least n" and "every connected components has size at least n" where n is a natural number, are not elusive for …
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at least~$2$, then $S$ is an end super …
A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a vertex or vertices in $D$. A graph $G$ is a $(2,2)$-dominated graph if it has a pair $(D,D')$ of disjoint $2$-dominating sets of vertices of $G$. In this paper we present two characterizations of minimal $(2,2)$-dominated graphs.
The learning complexity of special sets of vertices in graphs is studied in the model(s) of exact learning by (extended) equivalence and membership queries. Polynomial-time learning algorithms are described for vertex covers, independent sets, and dominating sets. The complexity of learning vertex sets of fixed size is also investigated, and it is shown that the k-element vertex covers in a graph can be learned in a number of rounds of interaction that is independent of the size of the graph. Apart from the elegance of these algorithmic problems, the chief motivation is the surprising recently…
The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and repres…
Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair o…
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. A dominating set $S$ is an isolate dominating set of $G$, if the induced subgraph $G[S]$ has at least one isolated vertex. The isolate domination number, $γ_0(G)$, is the minimum cardinality of an isolate dominating set of $G$. In this paper, we count the number of isolate dominating sets of some specific graphs.
A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. It is known that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs whose families of local maximum stable sets form greedoids have been analyzed as well. A unicycle graph owns only one cycle. In this paper we characterize the unicycle graphs whose families of local maximum stable sets form greedoids.
Galvin showed that for all fixed $δ$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $δ$ that admits the most independent sets is the complete bipartite graph $K_{δ,n-δ}$. He conjectured that except perhaps for some small values of $t$, the same graph yields the maximum count of independent sets of size $t$ for each possible $t$. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples $(n,δ, t)$ with $t\geq 3$, no $n$-vertex {\em bipartite} graph with minimum degree $δ$ admits more independent sets of size $t$ than…
Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$.The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$.For $k \geq 1$, a $k$-fair dominating set($kFD$-set) in $G$, is a dominating set $S$ such that $|N(v) \cap D|=k$ for every vertex $ v \in V\setminus D$.A fair dominating set, in $G$ is a $kFD$-set for some integer $k\geq 1$.In this paper, after presenting preliminaries, we count the number of fair dominating sets of some specific grap…
Let $T$ be a tree with a given adjacency eigenvalue $λ$. In this paper, by using the $λ$-minimal trees, we determine the structure of trees with a given multiplicity of the eigenvalue $λ$. Furthermore, we consider the relationship between the structure of trees and the eigensystem of a given Laplacian eigenvalue.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
A dominating set D of a graph G is a set such that each vertex v of G is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n-vertex plane triangulation has a dominating set of size at most n/3, and conjectured a bound of n/4 for n sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plummer and Zha (2009) and Honjo, Kawarabayashi, and Nakamoto (2009) extended the n/3 bound to triangulations on surfaces. We prove two related results: (i) There is a constant c such that any n-vertex plane triangulatio…
In a digraph $D = (X, \mathcal{U})$, not necessarily finite, an arc $(x, y) \in \mathcal{U}$ is reachable from a vertex $u$ if there exists a directed walk $W$ that originates from $u$ and contains $(x, y)$. A subset $S \subseteq X$ is an arc-reaching set of $D$ if for every arc $(x, y)$ there exists a diwalk $W$ originating at a vertex $u \in S$ and containing $(x, y)$. A minimal arc-reaching set is an arc-basis. $S$ is a point-reaching set if for every vertex $v$ there exists a diwalk $W$ to $v$ originating at a vertex $u \in S$. A minimal point-reaching set is a point-basis. We extend the r…
Let $α_{1} (G)$ denote the maximum size of an edge set that contains at most one edge from each triangle of $G$. Let $τ_{B} (G)$ denote the minimum size of an edge set whose deletion makes $G$ bipartite. It was conjectured by Lehel and independently by Puleo that $α_{1} (G) + τ_{B} (G) \le n^2/4$ for every $n$-vertex graph $G$. Puleo showed that $α_{1} (G) + τ_{B} (G) \le 5n^2/16$ for every $n$-vertex graph $G$. In this note, we improve the bound by showing that $α_{1} (G) + τ_{B} (G) \le 4403n^2/15000$ for every $n$-vertex graph $G$.
A ($κ$,$τ$)-regular set is a vertex subset S inducing a $κ$-regular subgraph such that every vertex out of S has $τ$ neighbors in S. This article is an expository overview of the main results obtained for graphs with ($κ$,$τ$)-regular sets. The graphs with classical combinatorial structures, like perfect matchings, Hamilton cycles, efficient dominating sets, etc, are characterized by ($κ$,$τ$)-regular sets whose determination is equivalent to the determination of those classical combinatorial structures. The characterization of graphs with these combinatorial structures are presented. The dete…
In a graph $Γ$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for …
In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for l…
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…
A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is the minimum cardinality of a semitotal dominating set of $G$ and is denoted by $γ_{t2}(G)$. In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs.
Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $°(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in B$, $°(u) \ge |B| - k - 1$. Moreover, we denote by $φ_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-small sets, and by $Ω_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-large sets. In this paper, we will show tight connections between $k$-small sets, respectively $k$-large sets, and the…
Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $δ_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} δ_S(v)\ge δ_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the …
A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for every $x \in S$. The fault-tolerant metric dimension (FTMD) $β'(G)$ of $G$ is the minimum cardinality of a fault-tolerant basis. It is shown that each twin vertex of $G$ belongs to every fault-tolerant basis of $G$. As a consequence, $β'(G) = n(G)$ iff each vertex of $G$ is a twin vertex, which corrects a wrong characterization of graphs $G$ with $β'(G) = n(G)…
We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_Γ$ is a quotient of Diestel's (denoted by $\vert G\vert_Θ$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite…
