The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals alpha(G) + mu(G), where mu(G) is the cardinality of a maximum matching in G. In this paper we characterize $α^{++}$-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a König-Egerváry graph is $α^{++}$-stable if and only if it has a perfect matching consisting of pendant edges and no four vertices of the graph span a cycle. As a corollary it gives necessary and sufficient conditions for $α^{++}$-stability of bipartite graphs and trees. For instance, we prove that a bipartite graph is $α^{++}$-stable if and only if it is well-covered and C4-free.
We consider $k$-cop-win graphs in the binomial random graph $G(n,1/2).$ It is known that almost all cop-win graphs contain a universal vertex. We generalize this result and prove that for every $k \in N$, almost all $k$-cop-win graphs contain a dominating set of cardinality $k$. From this it follows that the asymptotic number of labelled $k$-cop-win graphs of order $n$ is equal to $(1+o(1)) (1-2^{-k})^{-k} {n \choose k} 2^{n^2/2 - (1/2-\log_2(1-2^{-k})) n}$.
A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \in V(G)$, either $v \in D$ or $(v,d) \in E(G)$ for some $d \in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$ such that $|N[x] \cap S| \geq 2$. The cardinality redundance of $G$ is the minimum of $CR(S)$ taken over all dominating sets $S$. A set that achieves $CR(G)$ is a $γ_{cr}$-set, and the size of the minimum $γ_{cr}$-set is $γ_{cr}(G)$. We give the maximum number of edges in a graph with a given number of vertices and given cardinality redundance. In the cases t…
The stability number of a graph G, is the cardinality of a stable set of maximum size in G. If the stability number of G remains the same upon the addition of any edge, then G is called $α^{+}$-stable. G is a König-Egervary graph if its order equals the sum of its stability number and the cardinality of a maximum matching. In this paper we characterize $α^{+}$-stable König-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egervary graph is $α^{+}$-stable if and only if either the set of vertices belonging to no maximum stab…
In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets of complete bipartite graphs $K_{m,n}$ by solving the cases where $n=1$, $2$, $3$, $4$, and $5$, and also the case where $m=n$. In an article soon to be published, Krop, Minion, Patel, and Raridan concluded the edge-balanced index set problem for complete bipartite graphs with both parts of odd cardinality. In this paper, we conclude the problem for complete bipartite graphs where the larger part is of odd cardinality and the smaller is of even cardinality.
Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (M.Mahdian, On the computational complexity of strong edge coloring, Discrete Applied Mathematics, 118:239-248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an O(n^{11}) algorithm for the same (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in O(n^5) running time). In this paper we pro…
The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $α$-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect match…
A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the location-domination number $λ(G)$. An LD-set $S$ of a graph $G$ is global if it is an LD-set of both $G$ and its complement $\overline{G}$. The global location-domination number $λ_g(G)$ is the minimum cardinality of a global LD-set of $G$. In this work, we give some relations between locating-domina…
The $P_3$-hull number of a graph is the minimum cardinality of an infecting set of vertices that will eventually infect the entire graph under the rule that uninfected nodes become infected if two or more neighbors are infected. In this paper, we study the $P_3$-hull number for generalized Petersen graphs and a number of closely related graphs that arise from surgery or more generalized permutations. In addition, the number of components of the complement of an infecting set of minimum cardinality is calculated for the generalized Petersen graph and shown to always be $1$ or $2$. Moreover, inf…
We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the case $n\geq i+j$ and $q\geq 3$ we find the minimum cardinality of the support of such functions and obtain a characterization of functions with …
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 $G$ be a finite transitive group on a set $Ω$, let $α\in Ω$ and let $G_α$ be the stabilizer of the point $α$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{ω\in Ω\mid ω\textrm{ lies in a }G_α\textrm{-orbit of cardinality at most two}\}|}{|Ω|},$$ that is, the proportion of elements of $Ω$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5/6$, then each element of $Ω$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attai…
A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In this paper we provide a new characterization of bipartite graphs whose domination number is equal to the cardinality of its smaller partite set. Our characterization is based upon a new graph operation.
In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a dominating set. A set S subset V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is …
Let $G=(V,E)$ be a simple graph. For any integer $k\geq 1$, a subset of $V$ is called a $k$-tuple total dominating set of $G$ if every vertex in $V$ has at least $k$ neighbors in the set. The minimum cardinality of a minimal $k$-tuple total dominating set of $G$ is called the $k$-tuple total domination number of $G$. In this paper, we introduce the concept of upper $k$-tuple total domination number of $G$ as the maximum cardinality of a minimal $k$-tuple total dominating set of $G$, and study the problem of finding a minimal $k$-tuple total dominating set of maximum cardinality on several clas…
The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. If alpha(G-e) > alpha(G), then e is an alpha-critical edge, and if mu(G-e) < mu(G), then e is a mu-critical edge, where mu(G) is the cardinality of a maximum matching in G. G is a Koenig-Egervary graph if alpha(G) + mu(G) equals its order. Beineke, Harary and Plummer have shown that the set of alpha-critical edges of a bipartite graph is a matching. In this paper we generalize this statement to Koenig-Egervary graphs. We also prove that in a Koenig-Egervary graph alpha-critical edge…
The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or general position number) of G. In the paper, we obtain an upper bound and a lower bound regarding gp-number in all cactus with k cycles and t pendant edges. Furthermore, the gp-number of wheel graph is determined.
The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.
A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the \emph{location-domination number} $λ(G)$. An LD-set $S$ of a graph $G$ is \emph{global} if it is an LD-set of both $G$ and its complement $\overline{G}$. The \emph{global location-domination number} $λ_g(G)$ is the minimum cardinality of a global LD-set of $G$. For any LD-set $S$ of a given graph $G$…
A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $γ(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set edges $A\subseteq E(G)$ such that $γ(G-A)=γ(G)+1$. A chordal graph is a graph with no induced cycle of length four or more. In this paper, we prove that the bondage number of a chordal graph $G$ is at most the order of its maximum clique, that is, $b(G)\leq ω(G)$. We show that this bound is best possible.
By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the bitrades in the Hamming graph $H(n,k)$ with $k=3$ (ternary hypercube) and the growth of the number of such bitrades as $n$ grows. In particular, we determine all possible (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3}$) cardinatities of bitrades and prove that the cardinality of a bitrade is compartible to $0$ or $2^…
The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality alpha(G)=alpha(G^{2}), where G^{2} denotes the second power of G. In particular, we show that a Konig-Egervary graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-st…
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.
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $Π$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition of $V(G)$. Then $Π$ is called a {\it fixatic partition} if for all $i$; $1\leq i\leq k$, $F_i$ is a fixing set for $G$. The cardinality of a largest fixatic partition is called the {\it fixatic number} of $G$. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditi…
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 $G$. A dominating set $D$ is called co-even dominating set if the degree of vertex $v$ is even number for all $v\in V-D$. The cardinality of a smallest co-even dominating set of $G$, denoted by $γ_{coe}(G)$, is the co-even domination number of $G$. In this paper, we study the co-even domination number of some binary operations on graphs.
A matching $M$ in a graph $G$ is said to be uniquely restricted if there is no other matching in $G$ that matches the same set of vertices as $M$. We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching in an interval graph, thereby answering a question of Golumbic et al. ("Uniquely restricted matchings", M. C. Golumbic, T. Hirst and M. Lewenstein, Algorithmica, 31:139--154, 2001). Our algorithm actually solves the more general problem of computing a maximum cardinality "strong independent set" in an interval nest digraph, which may be of independe…
Let $p$ be a positive integer and $G=(V,E)$ be a simple graph. A subset $D\subseteq V$ is a $p$-dominating set if each vertex not in $D$ has at least $p$ neighbors in $D$. The $p$-domination number $\g_p(G)$ is the minimum cardinality among all $p$-dominating sets of $G$. The $p$-bondage number $b_p(G)$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with a $p$-domination number greater than the $p$-domination number of $G$. In this note we determine the 2-domination number $\g_2$ and 2-bondage number $b_2$ for the grid graphs $G_{m,n}=P_m\times P_n$ 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 $G$. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V-S$, there exists $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a smallest super dominating set of $G$, denoted by $γ_{sp}(G)$, is the super domination number of $G$. In this paper, we study super domination n…
A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $ρ^{\rm o}_L(G)$ and the largest cardinality is $ρ^{\rm o}(G)$. There exist graphs for which these two invariants are arbitrarily far apart. In this paper we begin the investigation of the class of graphs that have one size of maximal open packings. By presenting a method of constructing such graphs we show that every graph is the induced subgraph of a graph in this class. The main result of the paper is a…
For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ with minimum cardinality is called a basis of $G$ and its cardinality is the metric dimension of $G$. A connected graph $G$ is called randomly $k$-dimension…
Let $G$ be a simple and connected graph with vertex set $V(G)$. A vertex $w\in V(G)$ strongly resolves two vertices $u,v \in V(G)$ if $v$ belongs to a shortest $u-w$ path or $u$ belongs to a shortest $v-w$ path. A set $W \subseteq V(G)$ is a strong resolving set for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. A strong metric basis of $G$ is a strong resolving set for $G$ with minimum cardinality. The strong metric dimension of $G$, denoted by $sdim(G)$, is the cardinality of a strong metric basis of $G$. In this paper we compute the strong metric dimension …
For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from the closed neighborhood of $v$ are in $S.$ The domination number $γ(G)$ is the minimum cardinality of a dominating set of $G$, and the global offensive alliance number $γ_{o}(G)$ is the minimum cardinality of a global offensive alliance of $G$. We show that if $G$ is a connected unicycle graph of order $n$ with $l(G)$ leaves and $s(G)$ support vertices then $γ_…
A set D of vertices of a graph G=(V,E) is irredundant if each v of D satisfies (a) v is isolated in the subgraph induced by D, or (b) v is adjacent to a vertex in V-D that is nonadjacent to all other vertices in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the IR(G)-sets as vertex set, and sets D and D' are adjacent if and only if D' is obtained from D by exchanging a single vertex of D for an adjacent vertex in D'. We study the realizability of graphs as IR-graphs an…
Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is a $k$-metric generator for $G$ if for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. We study some problems regarding the complexity of some $k$-metric dimension problems. For instance, we show that the problem of computing the $k$-metric dimension of graphs is $NP$-Complete. However, the problem is solv…
The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has a labeling with $d$ labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs $G$ with infinite motion and growth $o \left(\frac{n^2}{\log_2 n} \right)$ is either $1$ or $2$, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs $G$ of arbitrary cardinality are $2$-dist…
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set in $G$, while the independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. We prove that for all integers $k \geq 3$ it holds that if $G$ is a connected $k$-regular graph, then $\frac{i(G)}{γ(G)} \leq \frac{k}{2}$, with equality if and only if $G = K…
A vertex $v$ is said to distinguish two other vertices $x$ and $y$ of a nontrivial connected graph G if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subseteq V(G)$ is a local metric set for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex of $S$. A local metric set with minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of $G$, denoted by $\dim_l(G)$. In this paper we present tight bounds for the local metric dimension of subgraph-amalgamation of graphs with special emphas…
A dominating set of a graph $G$ is a subset $D \subseteq V_G$ 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 $G$. The accurate domination number of $G$, denoted by $γ_{\rm a}(G)$, is the cardinality of a smallest set $D$ that is a dominating set of $G$ and no $|D|$-element subset of $V_G \setminus D$ is a dominating set of $G$. We study graphs for which the accurate domination number is equal to the domination number. In particular, all trees $G$ for which $γ_{\rm a…
A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. A set $C\subseteq V_G$ is a covering set of $G$ if every edge of $G$ has at least one vertex in $C$. The covering number $β(G)$ of $G$ is the minimum cardinality of a covering set of $G$. The set of connected graphs $G$ for which $γ(G)=β(G)$ is denoted by ${\cal C}_{γ=β}$, while ${\cal B}$ denotes the set of all connected bipartite graphs in which the domination numbe…
The stability number alpha(G) of a graph G is the cardinality of a maximum stable set in G, xi(G) denotes the size of core(G), where core(G) is the intersection of all maximum stable sets of G. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if xi(G)< 2, then G is quasi-regularizable; (ii) if G is of order n and alpha(G) > (n+k-1)/2, for some k > 0, then xi(G) > k, and xi(G) > k+1, whenever n+k-1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that alpha(G) > n/2 implies xi(G) > 0.…
A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted by $γ(G)$. A vertex set $D$ is a $k$-isolating set of $G$ if $G - N_{G}[D]$ contains no $k$-cliques. The minimum cardinality of a $k$-isolating set of $G$ is called the $k$-isolation number of $G$ and is denoted by $ι_{k}(G)$. Clearly, $γ(G) = ι_{1}(G)$. A vertex set $I$ is irredundant if, for every non-isolated vertex $v$ of $G[I]$, there exists a vertex $…
Let $G$ be a connected graph. Given an ordered set $W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$ and a vertex $u\in V(G)$, the representation of $u$ with respect to $W$ is the ordered $k$-tuple $(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$, where $d(u,w_i)$ denotes the distance between $u$ and $w_i$. The set $W$ is a metric generator for $G$ if every two different vertices of $G$ have distinct representations. A minimum cardinality metric generator is called a \emph{metric basis} of $G$ and its cardinality is called the \emph{metric dimension} of G. It is well known that the problem of finding the met…
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 called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V-S$, there exists $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. The cardinality of a smallest super dominating set of $G$, denoted by $γ_{sp}(G)$, is the super domination number of $G$. In this paper, we obtain more results on t…
Let $lct(G)$ be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2-connected graph $G$. We show that $lct(G)\leq k-1$ if $G$ is a partial $k$-tree and that $lct(G)\leq \max \{1, {ω(G){-}3}\}$ if $G$ is chordal, where $ω(G)$ is the cardinality of a maximum clique in $G$. Those results imply that all longest cycles intersect in 2-connected series parallel graphs and in 3-trees.
In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ for at most one for a sam…
In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $ψ(G)$, is the cardinality of a maximum dissociation set of $G$. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and…
A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $γ(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $γ$-$q$-{\it critical} if the smallest subset of edges whose subdivision necessarily increases $γ(G)$ has cardinality $q.$ In this paper we consider mainly $γ$-$q$-critical trees and give some general properties of $gamma$-$q$-critical graphs. In particular, we show that if $T$ is a $γ$-$q$-critical tree, then $1 \leq q \leq n(T)-1$ and we characterize extremal trees when $q=n(T)-1.$ Since a subdivisio…
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $α_c(G)$ of $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set of cardinality at least~$r$. The common independence number is squeezed between the independent domination number $i(G)$ and the independence number $α(G)$ of $G$, that is, $γ(G) \le i(G) \le α_c(G) \le α(G)$. A graph $G$ is dominat…
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number of components is restricted and illustrates a connection between graph compositions and Stirling numbers of the second kind.
Let $n$ be a positive integer, $σ$ be an element of the symmetric group $\mathcal{S}_n$ and let $σ$ be a cycle of length $n$. The elements $α,β\in \mathcal{S}_n$ are $σ$-equivalent, if there are natural numbers $k$ and $l$, such that $σ^k α=βσ^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $α= σ^{k_1} βσ^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph $Γ_n$ with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalenc…
For a connected graph $Γ=(V,E)$, a subset $R$ of ordered vertices in $V$ is said to be a resolving set in $Γ$, if the vector of distances to the vertices in $R$ is unique for each $u^{i}\in V(Γ)$. The metric dimension of $Γ$ is the minimum cardinality of such a set $R$. If $R\setminus \{u^{i}\}$ is still a resolving set $\forall$ $u^{i}\in R$, then $R$ is called a fault-tolerant resolving set (FTRS) for $Γ$ and its least cardinality is the fault-tolerant metric dimension (FTMD) of $Γ$. In this article, we introduce the concept of an independent fault-tolerant resolving set (IFTRS) and investig…
Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for ${\cal G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency…
Let $G$ be a graph. The Steiner distance of $W\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is necessarily a tree called a Steiner $W$-tree. The set $A\subseteq V(G)$ is a $k$-Steiner general position set if $V(T_B)\cap A = B$ holds for every set $B\subseteq A$ of cardinality $k$, and for every Steiner $B$-tree $T_B$. The $k$-Steiner general position number ${\rm sgp}_k(G)$ of $G$ is the cardinality of a largest $k$-Steiner general position set in $G$. Steiner cliques are introduced and used to bound ${\rm sgp}_k(G)$ from below. The $k$-Stei…
