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 established connection between the important unsolved problem of the learning complexity of CNF (or DNF) formulas and the learning complexity of dominating sets. The complexity of teaching sets of vertices in graphs is also considered.
Let $f$ be a $\{0,1\}$-valued function over an integer $d$-dimensional cube $\{0,1,\dots,n-1\}^d$, for $n \geq 2$ and $d \geq 1$. The function $f$ is called threshold if there exists a hyperplane which separates $0$-valued points from $1$-valued points. Let $C$ be a class of functions and $f \in C$. A point $x$ is essential for the function $f$ with respect to $C$ if there exists a function $g \in C$ such that $x$ is a unique point on which $f$ differs from $g$. A set of points $X$ is called teaching for the function $f$ with respect to $C$ if no function in $C \setminus \{f\}$ agrees with $f$…
A critical set in an $n \times n$ Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is $\lfloor n^2/4\rfloor$. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lo…
George Washington University（乔治华盛顿大学）；Lobachevsky State University（罗巴切夫斯基国立大学）；Higher School of Economics（高等经济学院）
It is known that a minimal teaching set of any threshold function on the twodimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them in the case of a minimal teaching set of size 3. We also prove that the average cardinality of the minimal teaching sets of threshold functions is asymptotically 7/2. We further present corollaries of these results concerning some special arrangements of lines in the plane.
We prove that every end of a graph contains either uncountably many disjoint rays or a set of disjoint rays that meet all rays of the end and start at any prescribed feasible set of start vertices. This confirms a conjecture of Georgakopoulos.
Strong cospectrality is an equivalence relation on the set of vertices of a graph that is of importance in the study of quantum state transfer in graphs. We construct families of abelian Cayley graphs in which the number of mutually strongly cospectral vertices can be arbitrarily large.
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.
In a graph G, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of maximum dissociation set problem in various subclasses of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and to no maximum dissociation set of a tree. We…
One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for trees of order greater than one, where, in fact, all the simplicial vertices are pendant. Namely, we show that any maximum stable set of such a tree contains at least one pendant vertex. Moreover, we prove that if T does not own a perfect matching, then a stable se…
We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n/4$, and no set of less than $\sqrt{n}/2$ vertices intersects all its maximum independent sets. This is motivated by a question of Bollobás, Erdős and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph $G$ on $n$ vertices with independence number $(1/4+\eps)n$, the average independence number of an induced subgraph of $G$ on a uniform random subset of the vertices is at most $(1/4+\eps…
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.
Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also pr…
An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by t…
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 $O(n^2 \log_2(n)^3)$.
Given a bipartite graph with bipartition $(A,B)$ where $B$ is equipartitioned into $k\ge2$ blocks, can the vertices in $A$ be picked one by one so that at every step, the picked vertices cover roughly the same number of vertices in each of these blocks? We show that, if each block has cardinality $m$, the vertices in $B$ have the same degree, and each vertex in $A$ has at most $cm$ neighbors in every block where $c>0$ is a small constant, then there is an ordering $v_1,\ldots,v_n$ of the vertices in $A$ such that for every $j\in\{1,\ldots,n\}$, the numbers of vertices with a neighbor in $\{v_1…
The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to \emph{percolate} if eventually all vertices are infected. For every $r \geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$. In the case $r=3$, for $n$ large enough, any graph on $n$ vertices with minimum degree $\lfloor n/2 \rfloor +1$ has a percolating set of size $3$ and for $r…
Two vertices u,v of connected graph G are doubly resolved by x,y\in V(G)if d(v; x)-d(u; x)\neq d(v; y)-d(u; y): A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. ψ(G) is the minimum cardinality of a doubly resolving set for the graph G. The aim of this paper is to investigate doubly resolving sets in graphs. An upper bound for Ψ(G) is obtained in terms of order and diameter of G. ψ(G) is computed for some important graphs and all graphs G of order n with the property ψ(G)=n-1 are characterized.
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 …
An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the $X$-sets of $G$ must satisfy certain conditions. Dominating sets, power dominating sets, zero forcing sets, and positive semidefinite zero forcing sets are all examples of $X$-sets. For graphs $G$ and $G'$ with no isolated vertices, it is shown that $G$ and $G'$ have isomorphic $X$-TAR reco…
Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $α(G)$ is the size of a maximum independent set in the graph. %An independent set with cardinality Let $R$ be a commutative ring with nonzero identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by $Γ(R)$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by $Γ_I(R)$, is the gr…
A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r\in R, then there exists s\in S so that S-\{s\}\cup \{r\} is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchang…
In the special case of graphs G of independence number a(G)=3 without induced chordless cycles C7 it is shown that exists connected dominating set D of vertices with number of vertices n(D)<=4. Using the concept of connected dominating sets, we defined a new invariant h(G) that does not exceed the number of Hadwiger. For the considered graphs it is shown that h(G)>=n(G)/4.
A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order n greater than 1, any stable set of size greater or equal to n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. Our main finding is the theorem claiming that if T does not own a perfect matching, then at least two pendant vertices an even distance apart belong to core(T). Wh…
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E (G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by some two vertices of $S$. The minimal doubly resolving set of vertices of graph $G$ is a doubly resolving set with minimum cardinality and is denoted by $ψ(G)$. In this paper, first, we construct a class of graphs of order $2n+ Σ_{r=1}^{k-2}nm^{r}$, denoted by $LSG(n,m, k)$, and call these graphs as the layer Sun graphs with parameters $n$, $m$ and $k$. More…
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 give a shorter proof of the recurrence relation for the domination polynomial $γ(P_{n},t)$ and for the number $γ_{k}(P_{n})$ of dominating $k$-sets of the path with $n$ vertices. For every positive integers $n$ and $k,$ numbers $γ_{k}(P_{n})$ are determined solving a problem posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating $k$-sets $γ_{k}(C_{n})$ of cycles and $γ_{k}(W_{n})$ of wheels with $n$ vertices are computed.
We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the $k$-colouring graph $\mathcal{C}_{k}(G)$ of a graph $G$ correspond to the proper $k$-colourings of a graph $G$, with two $k$-colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the $k$-edge-colouring graph $\mathcal{EC}_{k}(G)$ of $g$ are the proper $k$-edge-colourings of $G$, where two $k$-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two…
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $\partial(G)+\partial(\bar{G})\geq 2^n$ where $\partial(G)$ is the number of dominating sets in a graph $G$. In the same paper he comments that an upper bound for $\partial(G)+\partial(\bar{G})$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $\partial(G)+\par…
This papers focuses on the average order of dominating sets of a graph. We find the extremal graphs for the maximum and minimum value over all graphs on $n$ vertices, while for trees we prove that the star minimizes the average order of dominating sets. We prove the average order of dominating sets in graphs without isolated vertices is at most $3n/4$, but provide evidence that the actual upper bound is $2n/3$. Finally, we show that the normalized average, while dense in $[1/2,1]$, tends to $\frac{1}{2}$ for almost all graphs.
There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the Kruskal-Katona theorem that the lex graph has the maximum number of independent sets in a graph of fixed size and order. In this paper we solve two equivalent problems. The first is: what $3$-uniform hypergraph on a ground set of size $n$, having at least $t$ edges, has the most $2$-independent sets? Here a $2$--independent set is a subset of vertices contain…
For any integer $r\geqslant0$, the $r$-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least $r$ active neighbors becomes activated. A set of initially activated vertices leading to the activation of all vertices is said to be a percolating set. Denote the minimum size of a percolating set in the $r$-neighbor bootstrap percolation process on a graph $G$ by $m(G, r)$. In this paper, we present upper and lower bounds on $m(K_n^d, r)$, where $K_n^d$ is…
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…
In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. It is explained how these results can be used when studying partial difference sets in Abelian groups and projective two-weight sets. The underlying ideas are linear algebraic in nature.
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…
An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an analogue to the well-studied identifying code in the literature. In this paper, we prove that the optimal density of the OLD-set for the infinite triangular grid is $4/13$.
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.
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.
Let $S$ be a set of vertices of a graph $G$. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all but exactly one of its neighbors are in $cl(S)$, then the remaining neighbor is also in $cl(S)$. A set $S$ is called a zero forcing set of $G$ if $cl(S)=V(G)$. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a zero forcing set. Let $cl(N[S])$ be the set of vertices built from the closed neighborhood $N[S]$ of $S$, by iteratively applying the previous propagation rule. A set $S$ is called a power dominatin…
Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have $$\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$ where $v(H),v(G)$ and $e(H),e(G)$ denote the number of vertices and edges of the graph $H$ and $G$, respectively. In this paper we prove Sidorenko's conjecture for certain special graphs $G$: for the complete graph $K_q$ on $q$ vertices, for a $K_2$ with a loop added at one of the end vertices, and for a path on $3$ vertices with a loop added at ea…
The size of a largest independent set of vertices in a given graph $G$ is denoted by $α(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether $α(G) \geq K.$ An upper bound for the independence number $α(G)$ of a given graph $G$ with $n$ vertices and $m $ edges is given by $α(G) \leq p:=\lfloor\frac{1}{2} + \sqrt{\frac{1}{4} + n^2 - n - 2m}\rfloor.$ In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity $O(n…
Let $G$ be a Hamiltonian graph with $n$ vertices. A nonempty vertex set $X\subseteq V(G)$ is called a Hamiltonian cycle enforcing set (in short, an $H$-force set) of $G$ if every $X$-cycle of $G$ (i.e., a cycle of $G$ containing all vertices of $X$) is a Hamiltonian cycle. For the graph $G$, $h(G)$ is the smallest cardinality of an $H$-force set of $G$ and call it the $H$-force number of $G$. Ore's theorem states that the graph $G$ is Hamiltonian if $d(u)+d(v)\geq n$ for every pair of nonadjacent vertices $u,v$ of $G$. In this paper, we study the $H$-force sets of the graphs satisfying the con…
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.
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…
Given a $k$-colouring of the edges of the complete graph $K_n$, are there $k-1$ monochromatic components that cover its vertices? This important special case of the well-known Lovász-Ryser conjecture is still open. In this paper we consider a strengthening of this question, where we insist that the covering sets are not merely connected but have bounded diameter. In particular, we prove that for any colouring of $E(K_n)$ with 4 colours, there is a choice of sets $A_1, A_2, A_3$ that cover all vertices, and colours $c_1, c_2, c_3$, such that for each $i = 1,2,3$ the monochromatic subgraph induc…
Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved.
We study properties of 3-restricted min-wise independent sets of permutations and prove that the class of these sets is isomorphic to the class of subsets of vertices of multi-dimentional hypercube of certain size with certain pair-wise distances fixed.
