Search

Let $p \in \mathbb{N}$ and $q \in \mathbb{N} \cup \lbrace \infty \rbrace$. We study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of blue vertices, with all remaining vertices colored white. If a white vertex~$v$ has at least~$p$ blue neighbors and at least one of these blue neighbors of~$v$ has at most~$q$ white neighbors, then by the spreading color change rule the vertex~$v$ is recolored blue. The initial set $S$ of blue vertices is a $(p,q)$-spreading set for $G$ if by repeatedly applying the spreading color change rule all the vertices of $G$ are eventually colored blue. The $(p,q)$-spreading set is a generalization of the well-studied concepts of $k$-forcing and $r$-percolating sets in graphs. For $q \ge 2$, a $(1,q)$-spreading set is exactly a $q$-forcing set, and the $(1,1)$-spreading set is a $1$-forcing set (also called a zero forcing set), while for $q = \infty$, a $(p,\infty)$-spreading set is exactly a $p$-percolating set. The $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$ is the minimum cardinality of a $(p,q)$-spreading set. In this paper, we study $(p,q)$-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values $p$ and $q$ that are not both $1$ we either determine the $(p,q)$-spreading number of a claw-free cubic graph $G$ or show that $\sigma_{(p,q)}(G)$ attains one of two possible values.

A $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are $2$-packings. In this paper, we prove that every claw-free cubic graph admits a $(1,1,2,2)$-coloring. This implies that the conjecture from [Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph, Aequationes Math.\ 91 (2017) 169--184] that the packing chromatic number of subdivisions of subcubic graphs is at most $5$ is true in the case of claw-free cubic graphs., Comment: 12 pages, 2 figures, 17 references

Given a graph $G$ and a family of graphs $\cal F$, an $\cal F$-isolating set, as introduced by Caro and Hansberg, is any set $S\subset V(G)$ such that $G - N[S]$ contains no member of $\cal F$ as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an $\cal F$-isolating set in $G$. Following the domination games, Dominator (Staller) wants that the resulting $\cal F$-isolating set obtained at the end of the game, is as small (as big) as possible, which leads to the graph invariant called the game $\cal F$-isolation number, denoted $\iota_{\rm g}(G,\cal F)$. We prove that the Continuation Principle holds in the $\cal F$-isolation game, and that the difference between the game $\cal F$-isolation numbers when either Dominator or Staller starts the game is at most $1$. Considering two arbitrary families of graphs $\cal F$ and $\cal F'$, we find relations between them that ensure $\iota_{\rm g}(G,{\cal{F}}') \leq \iota_{\rm g}(G,{\cal{F}})$ for any graph $G$. A special focus is given on the isolation game, which takes place when ${\cal F}=\{K_2\}$. We prove that $\iota_{\rm g}(G,\{K_2\})\le |V(G)|/2$ for any graph $G$, and conjecture that $\lceil 3|V(G)|/7\rceil$ is the actual (sharp) upper bound. We prove that the isolation game on a forest when Dominator has the first move never lasts longer than the one in which Staller starts the game. Finally, we prove good lower and upper bounds on the game isolation numbers of paths $P_n$, which lead to the exact values $\iota_{\rm g}(P_n,\{K_2\})=\left\lfloor\frac{2n+2}{5}\right\rfloor$ when $n \equiv i \pmod 5$ and $i \in \{1,2,3\}$., Comment: 15 pages, 1 figure, 18 references

Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpi\'nski graphs and to give a short proof that Sierpi\'nski graphs are Class $1$. Sierpi\'nski-like graphs are also considered, including generalized Sierpi\'nski graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpi\'nski graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpi\'nski graphs and Kneser graphs $K(n, r)$ with $n \ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is not.

Given a graph $G$, the maximum size of an induced subgraph of $G$ each component of which is a star is called the edge open packing number, $\rho_{e}^{o}(G)$, of $G$. Similarly, the maximum size of an induced subgraph of $G$ each component of which is the star $K_{1,1}$ is the induced matching number, $\nu_I(G)$, of $G$. While the inequality $\rho_e^o(G)\geq \nu_{I}(G)$ clearly holds for all graphs $G$, we provide a structural characterization of those trees that attain the equality. We prove that the induced matching number of the lexicographic product $G\circ H$ of arbitrary two graphs $G$ and $H$ equals $\alpha(G)\nu_I(H)$. By similar techniques, we prove sharp lower and upper bounds on the edge open packing number of the lexicographic product of graphs, which in particular lead to NP-hardness results in triangular graphs for both invariants studied in this paper. For the direct product $G\times H$ of two graphs we provide lower bounds on $\nu_I(G\times H)$ and $\rho_{e}^{o}(G\times H)$, both of which are widely sharp. We also present sharp lower bounds for both invariants in the Cartesian and the strong product of two graphs. Finally, we consider the edge open packing number in hypercubes establishing the exact values of $\rho_e^o(Q_n)$ when $n$ is a power of $2$, and present a closed formula for the induced matching number of the rooted product of arbitrary two graphs over an arbitrary root vertex.

For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every $i\in\{1,\ldots,\ell\}$, is called an $S$-packing $\ell$-coloring of $G$. The minimum $\ell$ for which $G$ admits an $S$-packing $\ell$-coloring is called the $S$-packing chromatic number of $G$, denoted by $\chi_S(G)$. In this paper, we consider $S$-packing colorings of distance graphs $G(\mathbb{Z},\{k,t\})$, where $k$ and $t$ are positive integers, which are the graphs whose vertex set is $\mathbb{Z}$, and two vertices $x,y\in \mathbb{Z}$ are adjacent whenever $|x-y|\in\{k,t\}$. We complement partial results from two earlier papers, thus determining all values of $\chi_S(G(\mathbb{Z},\{k,t\}))$ when $S$ is any sequence with $s_i\le 2$ for all $i$. In particular, if $S=(1,1,2,2,\ldots)$, then the $S$-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while if $S=(1,2,2,\ldots)$, then the $S$-packing chromatic number is $5$, unless $\{k,t\}=\{2,3\}$ when it is $6$; when $S=(2,2,2,\ldots)$, the corresponding formula is more complex.

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G - S$ obtained by removing all vertices in $S$ is isolate-free. The domination number $\gamma(G)$ and the restrained domination number $\gamma_{r}(G)$ are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of $G$. Let $G$ be a cubic graph of order~$n$. A classical result of Reed [Combin. Probab. Comput. 5 (1996), 277--295] states that $\gamma(G) \le \frac{3}{8}n$, and this bound is best possible. To determine a best possible upper bound on the restrained domination number of $G$ is more challenging, and we prove that $\gamma_{r}(G) \le \frac{2}{5}n$., Comment: 39 pages, 16 figures

The $r$-neighbor bootstrap percolation is a graph infection process based on the update rule by which a vertex with $r$ infected neighbors becomes infected. We say that an initial set of infected vertices propagates if all vertices of a graph $G$ are eventually infected, and the minimum cardinality of such a set in $G$ is called the $r$-bootstrap percolation number, $m(G,r)$, of $G$. In this paper, we study percolating sets in direct products of graphs. While in general graphs there is no non-trivial upper bound on $m(G\times H,r)$, we prove several upper bounds under the assumption $\delta(G)\ge r$. We also characterize the connected graphs $G$ and $H$ with minimum degree $2$ that satisfy $m(G \times H, 2) = \frac{|V(G \times H)|}{2}$. In addition, we determine the exact values of $m(P_n \times P_m, 2)$, which are $m+n-1$ if $m$ and $n$ are of different parities, and $m+n$ otherwise., Comment: 21 pages, 6 figures

Given a graph $G$, two edges $e_{1},e_{2}\in E(G)$ are said to have a common edge $e$ if $e$ joins an endvertex of $e_{1}$ to an endvertex of $e_{2}$. A subset $B\subseteq E(G)$ is an edge open packing set in $G$ if no two edges of $B$ have a common edge in $G$, and the maximum cardinality of such a set in $G$ is called the edge open packing number, $\rho_{e}^{o}(G)$, of $G$. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree $4$, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper [Edge open packing sets in graphs, RAIRO-Oper.\ Res.\ 56 (2022) 3765--3776]. Notably, we characterize the graphs $G$ that attain the upper bound $\rho_e^o(G)\le |E(G)|/\delta(G)$, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result., Comment: 12 pages, 1 figure

In this follow-up to [M.G.~Cornet, P.~Torres, arXiv:2308.15603], where the $k$-tuple domination number and the 2-packing number in Kneser graphs $K(n,r)$ were studied, we are concerned with two variations, the $k$-domination number, ${\gamma_{k}}(K(n,r))$, and the $k$-tuple total domination number, ${\gamma_{t\times k}}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that ${\gamma_{k}}(K(n,r))\ge {\gamma_{k}}(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and ${\gamma_{t\times k}}(K(n,r))\ge {\gamma_{t\times k}}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that ${\gamma_{k}}(K(n,r))={\gamma_{t\times k}}(K(n,r))=k+r$ when $n\geq r(k+r)$, and that in this case every ${\gamma_{k}}$-set and ${\gamma_{t\times k}}$-set is a clique, while ${\gamma_{k}}(r(k+r)-1,r)={\gamma_{t\times k}}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the 2-packing number, $\rho_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $\rho_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $\rho_2(K(n,r))$ to be equal to some small values by imposing bounds on $r$ with respect to $n$. We also prove a version of monotonicity for the $2$-packing number of Kneser graphs., Comment: 15 pages, 3 tables

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$ by two players, Dominator and Staller, where Dominator wants to finish the game in as few rounds as possible while Staller wants just the opposite. In each round, Dominator indicates a vertex $u$ of $G$ that has not been dominated by previous selections of Staller, which, by the rules of the game, forces Staller to select a vertex in the closed neighborhood of $u$. The game is finished when all vertices of $G$ become dominated by the vertices selected by Staller. Assuming that both players are playing optimally according to their goals, the number of selected vertices during the game is the indicated domination number, $\gamma_i(G)$, of $G$. We prove several bounds on the indicated domination number expressed in terms of other graph invariants. In particular, we find a place of the new graph invariant in the well-known domination chain, by showing that $\gamma_i(G)\ge \Gamma(G)$ for all graphs $G$, and by showing that the indicated domination number is incomparable with the game domination number and also with the upper irredundance number. In connection with the trivial upper bound $\gamma_i(G)\le n(G)-\delta(G)$, we characterize the class of graphs $G$ attaining the bound provided that $n(G)\ge 2\delta(G)+2$. We prove that in trees, split graphs and grids the indicated domination number equals the independence number. We also find a formula for the indicated domination number of powers of paths, from which we derive that there exist graphs in which the indicated domination number is arbitrarily larger than the upper irredundance number., Comment: 19 pages

A sequence $S$ of vertices of a graph $G$ is called a dominating sequence of $G$ if $(i)$ each vertex $v$ of $S$ dominates a vertex of $G$ that was not dominated by any of the vertices preceding vertex $v$ in $S$, and $(ii)$ every vertex of $G$ is dominated by at least one vertex of $S$. The Grundy Domination problem is to find a longest dominating sequence for a given graph $G$. It has been known that the decision version of the Grundy Domination problem is NP-complete even when restricted to chordal graphs. In this paper, we prove that the decision version of the Grundy Domination problem is NP-complete for bipartite graphs and co-bipartite graphs. On the positive side, we present a linear-time algorithm that solves the Grundy Domination problem for chain graphs, which form a subclass of bipartite graphs., Comment: 18 pages, 3 figures

While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number $\gamma_t(G)$ (resp. $\Gamma_t(G)$) of $G$. We prove that $Z(G)+\gamma_t(G)\le n(G)$ and $Z(G)+\frac{\Gamma_t(G)}{2}\le n(G)$ holds for any graph $G$ with no isolated vertices of order $n(G)$. Both bounds are sharp as demonstrated by several infinite families of graphs. In particular, we show that every graph $H$ is an induced subgraph of a graph $G$ with $Z(G)+\frac{\Gamma_t(G)}{2}=n(G)$. Furthermore, we prove a characterization of graphs with power domination equal to $1$, from which we derive a characterization of the extremal graphs attaining the trivial lower bound $Z(G)\ge \delta(G)$. The class of graphs that appears in the corresponding characterizations is obtained by extending an idea from [D.D.~Row, A technique for computing the zero forcing number of a graph with a cut-vertex, Linear Alg.\ Appl.\ 436 (2012) 4423--4432], where the graphs with zero forcing number equal to $2$ were characterized., Comment: 18 pages, 1 figure

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph $G$ are contaminated initially, before the process starts. By the $q$-forcing rule, a contaminated vertex having at most $q$ uncontaminated neighbors enforces all the neighbors to become contaminated, while by the $p$-percolation rule, an uncontaminated vertex becomes contaminated if at least $p$ of its neighbors are contaminated. In this paper, we consider sets $S$ that are at the same time $q$-forcing sets and $p$-percolating sets, and call them $(p,q)$-spreading sets. Given positive integers $p$ and $q$, the minimum cardinality of a $(p,q)$-spreading set in $G$ is a $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$. While $q$-forcing sets have been studied in a dozen of papers, the decision version of the corresponding graph invariant has not been considered earlier, and we fill the gap by proving its NP-completeness. This, in turn, enables us to prove the NP-completeness of the decision version of the $(p,q)$-spreading number in graphs for an arbitrary choice of $p$ and $q$. Again, for every $p\in \mathbb{N}$ and $q\in\mathbb{N}\cup\{\infty\}$, we find a linear-time algorithm for determining the $(p,q)$-spreading number of a tree. In addition, we present a lower and an upper bound on the $(p,q)$-spreading number of a tree and characterize extremal families of trees. In the case of square grids, we combine some known results and new results on $(2,1)$-spreading and $(4,q)$-spreading to obtain $\sigma_{(p,q)}(P_m\Box P_n)$ for all $(p,q)\in (\mathbb{N}\setminus\{3\})\times (\mathbb{N}\cup\{\infty\})$ and all $m,n\in\mathbb{N}$., Comment: 20 pages, 5 figures

The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp. $2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which intersect. It is proved that if $G$ is bipartite, then $\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$. For hypercubes, the lower bounds $\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$ and $\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$ are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest hypercube which is not perfect injectively colorable. It is also proved that $\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$, where $H$ is an arbitrary graph with no isolated vertices., Comment: 11 pages, 2 figures, 1 table

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths., Comment: 23 pages

Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every two vertices in $X$ (respectively, in $G$) there is a shortest path whose internal vertices are not in $X$ is a mutual-visibility (respectively, total mutual-visibility) set in $G$. The cardinality of a largest (total) mutual-visibility set in $G$ is known under the name (total) mutual-visibility number, and has been studied in several recent works. In this paper, we propose two lower variants of the mentioned concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in $G$, and denote them by $\mu^{-}(G)$ and $\mu_t^{-}(G)$, respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph $G$, we prove that both differences $\mu^{-}(G)-\mu_t^{-}(G)$ and $\mu_t^{-}(G)-\mu^{-}(G)$ can be arbitrarily large. We characterize graphs $G$ with some small values of $\mu^{-}(G)$ and $\mu_t^{-}(G)$, and prove a useful tool called Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with Bollob\'{a}s-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to $\mu_t^{-}(G)$., Comment: 21 pages, 3 figures

The Grundy domination number, ${\gamma_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j

The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377].

Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of $G$, ${\gpack}(G)$, is the maximum cardinality of a geodesic packing in $G$. It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, ${\gt}(G)$, which is the minimum cardinality of a set of vertices that hit all maximal geodesics in $G$. While $\gt(G)\ge \gpack(G)$ in every graph $G$, the quotient ${\rm gt}(G)/{\rm gpack}(G)$ is investigated. By using the rook's graph, it is proved that there does not exist a constant $C < 3$ such that $\frac{{\rm gt}(G)}{{\rm gpack}(G)}\le C$ would hold for all graphs $G$. If $T$ is a tree, then it is proved that ${\rm gpack}(T) = {\rm gt}(T)$, and a linear algorithm for determining ${\rm gpack}(T)$ is derived. The geodesic packing number is also determined for the strong product of paths.

A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. We determine a smallest geodesic transversal in certain interconnection networks such as mesh of trees, and some well-known chemical structures such as silicate networks and carbon nanosheets. Some useful general bounds for the corresponding graph invariant are obtained along the way., Comment: 12 Pages and 9 Figures

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we study dissociation in the well-known class of Kneser graphs $K_{n,k}$. In particular, we establish that the dissociation number of Kneser graphs $K_{n,2}$ equals $\max{\{n-1,6\}}$. We show that for any $k \geq 2$, there exists $n_0 \in \mathbb{N}$ such that $diss(K_{n,k})=\alpha(K_{n,k})$ for any $n \geq n_0$. We consider the case $k=3$ in more details and prove that $n_0=8$ in this case. Then we improve a trivial upper bound $2\alpha(K_{n,k})$ for the dissociation number of Kneser graphs $K_{n,k}$ by using Katona's cyclic arrangement of integers from $\{1,\ldots , n\}$. Finally we investigate the odd graphs, that is, the Kneser graphs with $n=2k+1$. We prove that $diss(K_{2k+1,k})={2k \choose k}$., Comment: 9 pages, 1 figure

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $\chi'(G \setminus F)<\chi'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{\chi'}(G)$, and such a (smallest) set $F$ is called a (minimum) mitigating set. While $1\le es_{\chi'}(G)\le \lfloor n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal and near-extremal values of $es_{\chi'}(G)$. The graphs $G$ with $es_{\chi'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with $es_{\chi'}(G)=\lfloor n/2\rfloor-1$ and $\chi'(G)=\Delta(G)+1$ are characterized. We establish that the odd cycles and $K_2$ are exactly the regular connected graphs with the chromatic edge stability index $1$; on the other hand, we prove that it is NP-hard to verify whether a graph $G$ has $es_{\chi'}(G)=1$. We also prove that every minimum mitigating set of an $r$-regular graph $G$, where $r\ne 4$, with $es_{\chi'}(G)=2$ is a matching. Furthermore, we propose a conjecture that for every graph $G$ there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs $G$ with $es_{\chi'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$, and for bipartite graphs., Comment: 17 pages

An open packing in a graph $G$ is a set $S$ of vertices in $G$ such that no two vertices in $S$ have a common neighbor in $G$. The injective chromatic number $\chi_i(G)$ of $G$ is the smallest number of colors assigned to vertices of $G$ such that each color class is an open packing. Alternatively, the injective chromatic number of $G$ is the chromatic number of the two-step graph of $G$, which is the graph with the same vertex set as $G$ in which two vertices are adjacent if they have a common neighbor. The concept of injective coloring has been studied by many authors, while in the present paper we approach it from two novel perspectives, related to open packings and the two-step graph operation. We prove several general bounds on the injective chromatic number expressed in terms of the open packing number. In particular, we prove that $\chi_{i}(G)\geq \frac{1}{2}+\sqrt{\frac{1}{4}+\frac{2m-n}{\opack}}$ holds for any connected graph $G$ of order $n\geq2$, size $m$, and the open packing number $\opack$, and characterize the class of graphs attaining the bound. Regarding the well-known bound $\chi_i(G)\ge \Delta(G)$, we describe the family of extremal graphs and prove that deciding when the equality holds (even for regular graphs) is NP-complete, solving an open problem from an earlier paper. Next, we consider the chromatic number of the two-step graph of a graph, and compare it with the clique number and the maximum degree of the graph. We present two large families of graphs in which $\chi_i(G)$ equals the cardinality of a largest clique of the two-step graph of $G$. Finally, we consider classes of graphs that admit an injective coloring in which all color classes are maximal open packings. We give characterizations of three subclasses of these graphs among graphs with diameter $2$, and find a partial characterization of hypercubes with this property.

We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph $G$, and Alice's goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round $i$, where $i=1,2,,\ldots$, the players are taking turns in selecting a previously unselected vertex of $G$ and giving it color $i$ (hence, in each round the selected vertices form an independent set). The game ends when all vertices of $G$ are selected (and thus colored), and the total number of rounds during the game when both players are playing optimally with respect to their goals, is called the independence game chromatic number, $\chi_{ig}(G)$, of $G$. In fact, four different versions of the independence game chromatic number are considered, which depend on who starts a game and who starts next rounds. We prove that the new invariants lie between the chromatic number of a graph and the maximum degree plus $1$, and characterize the graphs in which each of the four versions of the game invariant equals $2$. We compare the versions of the independence game chromatic number among themselves and with the classical game chromatic number. In addition, we prove that the independence game chromatic number of a tree can be arbitrarily large.

The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph $G$ starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set $S\subset V(G)$ is called a zero forcing set of $G$ if initially only vertices of $S$ are active and the described process enforces all vertices of $G$ to become active. The size of a minimum zero forcing set in $G$ is called the zero forcing number of $G$. While a minimum zero forcing set can only be unique in edgeless graphs, we consider the weaker uniqueness condition, notably that for every two minimum zero forcing sets in a graph $G$ there is an automorphism that maps one to the other. We characterize the class of trees that enjoy this condition by using properties of minimum path covers of trees. In addition, we investigate both variations of uniqueness for several concepts of Grundy domination, which first appeared in the context of domination games, yet they are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests., Comment: 22 pages

A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph $G$ is introduced as the task to find a smallest set $S$ of vertices of $G$ such that each maximal geodesic has at least one vertex in $S$. The minimum cardinality of such a set is the geodesic-transversal number ${\rm gt}(G)$ of $G$. It is proved that ${\rm gt}(G) = 1$ if and only if $G$ is a subdivided star and that the geodesic-transversal problem is NP-complete. Fast algorithms to determine the geodesic-transversal number of trees and of spread cactus graphs are designed, respectively.

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number, $\gamma_{\rm gr}(G)$, of $G$. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $\gamma_{\rm gr}(G) \geq \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$ holds for every connected $k$-regular graph of order $n$ different from $K_{k+1}$ and $\bar{2C_4}$. The bound in the case $k=3$ reduces to $\gamma_{\rm gr}(G) \geq \frac{n}{2}$, and we characterize the connected cubic graphs with $\gamma_{\rm gr}(G)=\frac{n}{2}$. If $G$ is different from $K_4$ and $K_{3,3}$, then $\frac{n}{2}$ is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $\gamma(D)$ (respectively,$\gamma_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $\rho(D)$ (respectively,$\rho^{\rm o}(D)$). We prove that in digraphs whose underlying graphs have girth at least $7$, the closed (respectively,open) in-neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree $T$ (that is, a digraph whose underlying graph is a tree), $\gamma_t(T)=\rho^{\rm o}(T)$ and $\gamma(T)=\rho(T)$. By using the former equality we then prove that $\gamma_t(G\times T)=\gamma_t(G)\gamma_t(T)$, where $G$ is any digraph and $T$ is any ditree, each without a source vertex, and $G\times T$ is their direct product. From the equality $\gamma(T)=\rho(T)$ we derive the bound $\gamma(G\mathbin{\Box} T)\ge\gamma(G)\gamma(T)$, where $G$ is an arbitrary digraph, $T$ an arbitrary ditree and $G\mathbin{\Box} T$ is their Cartesian product. In general digraphs this Vizing-type bound fails, yet we prove that for any digraphs $G$ and $H$, where $\gamma(G)\ge\gamma(H)$, we have $\gamma(G \mathbin{\Box} H) \ge \frac{1}{2}\gamma(G)(\gamma(H) + 1)$. This inequality is sharp as demonstrated by an infinite family of examples. Ditrees $T$ and digraphs $H$ enjoying $\gamma(T\mathbin{\Box} H)=\gamma(T)\gamma(H)$ are also investigated., Comment: 22 pages and 5 figures

Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$ the distance between $u$ and $v$ in $G$ is greater than $a_i$. The smallest $k$ such that $G$ has an $S$-packing $k$-coloring is the $S$-packing chromatic number, $\chi_S(G)$, of $G$. In this paper, we consider the distance graphs $G(\mathbb{Z},\{2,t\})$, where $t>1$ is an odd integer, which has $\mathbb{Z}$ as its vertex set, and $i,j\in\mathbb{Z}$ are adjacent if $|i-j|\in\{2,t\}$. We determine the $S$-packing chromatic numbers of the graphs $G(\mathbb{Z},\{2,t\})$, where $S$ is any sequence with $a_i\in\{1,2\}$ for all $i$. In addition, we give lower and upper bounds for the $d$-distance chromatic numbers of the distance graphs $G(\mathbb{Z},\{2,t\})$, which in the cases $d\ge t-3$ give the exact values. Implications for the corresponding $S$-packing chromatic numbers of the circulant graphs are also discussed., Comment: 21 pages, 3 figures

The chromatic edge-stability number ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $\chi(G')=\chi(G)-1$. Edge-stability critical graphs are introduced as the graphs $G$ with the property that ${\rm es}_{\chi}(G-e) < {\rm es}_{\chi}(G)$ holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph with $\chi(G)=k$ and ${\rm es}_{\chi}(G)=\ell$, then $G$ is $(k,\ell)$-critical. Graphs which are $(3,2)$-critical and contain at most four odd cycles are classified. It is also proved that the problem of deciding whether a graph $G$ has $\chi(G)=k$ and is critical for the chromatic number can be reduced in polynomial time to the problem of deciding whether a graph is $(k,2)$-critical.

Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies that vertices $u$ and $v$ are at distance greater than $i$, is called a packing coloring of $G$. The minimum number of colors in a packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $\chi_\rho(G)$. In this paper, we propose the study of $\chi_\rho$-critical graphs, which are the graphs $G$ such that for any proper subgraph $H$ of $G$, $\chi_\rho(H)<\chi_\rho(G)$. We characterize $\chi_\rho$-critical graphs with diameter 2, and $\chi_\rho$-critical block graphs with diameter 3. Furthermore, we characterize $\chi_\rho$-critical graphs with small packing chromatic numbers, and we also consider $\chi_\rho$-critical trees. In addition, we prove that for any graph $G$ with $e\in E(G)$, we have $(\chi_\rho(G)+1)/2\le \chi_\rho(G-e)\le \chi_\rho(G)$, and provide a corresponding realization result, which shows that $\chi_\rho(G-e)$ can achieve any of the integers between the bounds., Comment: 19 pages, 1 figure

A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from $P_2$. We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least $6$. Finally, we show that the recognition of the graphs with a unique maximum open packing is polynomially equivalent to the recognition of the graphs with a unique maximum independent set, and we prove that the complexity of both problems is not polynomial, unless P=NP., Comment: 16 pages, 4 figures