arXiv subjects

Tanja Dravec

Publications and source records attributed to Tanja Dravec.

At least 19 recordsLinked to original sources

arXiv2026

Domination game and total domination game played on Sierpiński graphs

The game domination numbers $γ_{\rm g}$ and $γ_{\rm g}^\prime$, and the game total domination numbers $γ_{\rm{tg}}$ and $γ_{\rm{tg}}^\prime$ are investigated on Sierpiński graphs $S_p^n$. For the domination game, the trivial bounds arising from the known domination number and the Grundy domination number of $S_p^n$ are significantly improved by proving that $(2p-3)p^{n-2}\leq γ_{\rm g}(S_p^n), γ_{\rm g}^\prime(S_p^n) \leq (2p-2)p^{n-2}$. For the total domination game the bounds $γ_{\rm{tg}}(S_p^n), γ_{\rm{tg}}^\prime(S_p^n) \geq (2p-2)p^{n-2}$ are established.

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arXiv2026

On the Independence Number of the Modular Product

The \emph{modular product} $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. The independence number $α(G)$ of a graph $G$ is the maximum cardinality of a set of pairwise nonadjacent vertices in $G$. In this paper, we study the independence number of the modular product of graphs. We first structurally characterize all independent set of $G\diamond H$ which lead to the exact result on $α(G \diamond H)$. Special cases of this result lead to several sharp bounds and some exact results for $α(G \diamond H)$. Finally, we introduce a partition graph associated with $G \diamond H$ that provides a framework for constructing independent sets of the modular product from independent sets of its substructures.

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arXiv2026

Bounds on the game isolation number and exact values for paths and cycles

The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $ι_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $ι_{\rm g}'(G)$. In this paper, $ι_{\rm g}(C_n)$, $ι_{\rm g}(P_n)$, $ι_{\rm g}'(C_n)$, and $ι_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $ι_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $ι_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $ι_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $ι_{\rm g}(G) = ι_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds.

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arXiv2025

Thresholds for the biased Maker-Breaker domination games

In the $(a,b)$-biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph $G$ such that Dominator selects $a$ and Staller selects $b$ vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of $G$, while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer $b$, Dominator's threshold, $\textrm{a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, $\textrm{a}'_b$ denotes the minimum $a$ such that Dominator wins when Staller starts the $(a,b)$-biased game. Staller's thresholds, $\textrm{b}_a$ and $\textrm{b}'_a$, are defined analogously. It is proved that Staller wins the $(k-1,k)$-biased games in a graph $G$ if its order is sufficiently large with respect to a function of $k$ and the maximum degree of $G$. Along the way, the $\ell$-local domination number of a graph is introduced. This new parameter is proved to bound Dominator's thresholds $\textrm{a}_\ell$ and $\textrm{a}_\ell'$ from above. As a consequence, $\textrm{a}_1'(G)\le 2$ holds for every claw-free graph $G$. More specific results are obtained for thresholds in line graphs and Cartesian grids. Based on the concept of $[1,k]$-factor of a graph $G$, we introduce the star partition width $σ(G)$ of $G$, and prove that $\textrm{a}_1'(G)\le σ(G)$ holds for any nontrivial graph $G$, while $\textrm{a}_1'(G)=σ(G)$ if $G$ is a tree.

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arXiv2025

A proof of the $\frac{3}{8}$-conjecture for independent domination in cubic graphs

A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in~$S$. An independent dominating set in $G$ is a dominating set of $G$ with the additional property that it is an independent set. The domination number, $γ(G)$, and the independent domination number, $i(G)$, are the minimum cardinalities among all dominating sets and independent dominating sets in $G$, respectively. By definition, $γ(G) \le i(G)$ for all graphs $G$. Let $G$ be a connected cubic graph of order~$n$. In 1996 Reed [Combin.\ Probab.\ Comput.\ 5 (1996), 277--295] proved a breakthrough result that $γ(G) \le \frac{3}{8}n$. We prove the stronger result that if $G$ is different from $K_{3,3}$ and the $5$-prism $C_5 \, \Box \, K_2$, then $i(G) \le \frac{3}{8}n$. This proves a known conjecture. The bound is tight in the sense that there are infinite families of connected cubic graphs that achieve equality in this bound.

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arXiv2025

Monophonic number of Kneser graphs and strongly 2-monophonic graphs

Given a graph $G$ a set $S\subset V(G)$ is called monophonic if every vertex in $G$ lies on some induced path between two vertices in $S$. The monophonic number, $m(G)$, of $G$, which is the smallest cardinality of a monophonic set in $G$, has been studied from various perspectives. In this paper, we establish $m(K(n,r))$ for all Kneser graphs $K(n,r)$, where $n\ge 2r$. In addition, when $r\ge 3$, we prove an even stronger property, notably that every pair of non-adjacent vertices in $K(n,r)$ forms a monophonic set. We call the graphs satisfying this property strongly $2$-monophonic graphs. We present several (sufficient and necessary) conditions for a graph to be strongly $2$-monophonic, and prove that the Cartesian product of any two strongly $2$-monophonic graphs is also such. Besides non-complete Hamming graphs, we also prove that every Johnson graph is strongly $2$-monophonic, whereas chordal graphs, with the exception of the graphs $K_n-e$, do not enjoy this property.

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arXiv2025

Isolation number: Cartesian and lexicographic products and generalized Sierpiński graphs

The isolation number $ι(G)$ of a graph $G$ is the minimum cardinality of a set $A\subset V(G)$ such that the subgraph induced by the vertices that are not in the union of the closed neighborhoods of vertices in $A$ has no edges. The invariant, known also under the name vertex-edge domination number of $G$, has attracted a lot of interest in recent years. In this paper, we study the behavior of the isolation number under several graph operations, namely the Cartesian and the lexicographic product and the fractalization leading to generalized Sierpiński graphs. We prove several upper and lower bounds on the isolation number of the Cartesian product of two graphs. We prove a lower bound for the isolation number of the prism $G\,\Box\, K_2$ over an arbitrary graph $G$, which in the case of bipartite graphs leads to the equality $ι(G\,\Box\, K_2)=γ(G)$, where $γ(G)$ is the domination number of $G$. In particular, $ι(Q_{n+1})=γ(Q_n)$ holds for all positive integers $n$, where $Q_n$ is the $n$-dimensional hypercube. For the lexicographic product $G\circ H$ we prove that its isolation number, under certain mild restrictions, equals the total domination number of the first factor $G$. We also prove sharp lower and upper bounds on the isolation numbers of the generalized Sierpiński graphs $S_G^t$, where $G$ is an arbitrary base graph. These bounds in the case of classical Sierpiński graphs, namely $S_{K_n}^t$, coincide and lead to the exact values $ι(S_{K_n}^t)=(n-1)\cdot n^{t-2}$ for all dimensions $t\ge 2$.

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arXiv2025

On Maker-Breaker domination game critical graphs

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller who alternate turns selecting an unplayed vertex of $G$. The goal of Dominator is that the vertices he selected during the game form a dominating set while Staller's goal is to prevent this from happening. The graph invariant $γ_{\rm MB}'(G)$ is the number of Dominator's moves in the game played on $G$ in which he can achieve his goal when Staller makes the first move and both players play optimally. In this paper, we continue the investigation of $2$-$γ_{\rm MB}'$-critical graphs, initiated in [Divarakan et al., Maker--Breaker domination game critical graphs, Discrete Appl.\ Math. 368 (2025) 126--134], which are defined as the graphs $G$ with $γ_{\rm MB}'(G)=2$ and $γ_{\rm MB}'(G-e)>2$ for every edge $e$ in $G$. The authors characterized bipartite $2$-$γ_{\rm MB}'$-critical graphs, and found an example of a non-bipartite $2$-$γ_{\rm MB}'$-critical graph. In this paper, we characterize the $2$-$γ_{\rm MB}'$-critical graphs that have a cut-vertex, which are represented by two infinite families. In addition, we prove that $C_5$ is the only non-bipartite, triangle-free $2$-$γ_{\rm MB}'$-critical graph.

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arXiv2025

Graphs with unique Grundy dominating sets

Given a graph $G$ consider a procedure of building a dominating set $D$ in $G$ by adding vertices to $D$ one at a time in such a way that whenever vertex $x$ is added to $D$ there exists a vertex $y\in N_G[x]$ that becomes dominated only after $x$ is added to $D$. The maximum cardinality of a set $D$ obtained in the described way is called the Grundy domination number of $G$ and $D$ a Grundy dominating set. While a Grundy dominating set of a connected graph $G$ is not unique unless $G$ is the trivial graph, we consider a natural weaker uniqueness condition, notably that for every two Grundy dominating sets in a graph $G$ there is an automorphism that maps one to the other. We investigate both versions of uniqueness for several concepts of Grundy domination, which appeared in the context of domination games and 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.

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arXiv2025

Maker-Breaker domination game critical graphs

The Maker-Breaker domination game (MBD game) is a two-player game played on a graph $G$ by Dominator and Staller. They alternately select unplayed vertices of $G$. The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.

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arXiv2025

The radius capture number

In the classic cop and robber game, two players--the cop and the robber--take turns moving to a neighboring vertex or staying at their current position. The cop aims to capture the robber, while the robber tries to evade capture. A graph $G$ is called a cop-win graph if the cop can always capture the robber in a finite number of moves. In the cop and robber game with radius of capture $k$, the cop wins if he can come within distance $k$ of the robber. The radius capture number $\rc(G)$ of a graph $G$ is the smallest $k$ for which the cop has a winning strategy in this variant of the game. In this paper, we establish that $\rc(H) \leq \rc(G)$ for any retract $H$ of $G$. We derive sharp upper and lower bounds for the radius capture number in terms of the graph's radius and girth, respectively. Additionally, we investigate the radius capture number in vertex-transitive graphs and identify several families $\cal{F}$ of vertex-transitive graphs with $\rc(G)=\rad(G)-1$ for any $G \in \cal{F}$. We further study the radius capture number in outerplanar graphs, Sierpiński graphs, harmonic even graphs, and graph products. Specifically, we show that for any outerplanar graph $G$, $\rc(G)$ depends on the size of its largest inner face. For harmonic even graphs and Sierpiński graphs $S(n,3)$, we prove that $\rc(G)=\rad(G)-1$. Regarding graph products, we determine exact values of the radius capture number for strong and lexicographic products, showing that they depend on the radius capture numbers of their factors. Lastly, we establish both lower and upper bounds for the radius capture number of the Cartesian product of two graphs.

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arXiv2025

On the stress transit function

The stress interval $S(u,v)$ between $u,v\in V(G)$ is the set of all vertices in a graph $G$ that lie on every shortest $u,v$-path. A set $U \subseteq V(G)$ is stress convex if $S(u,v) \subseteq U$ for any $u,v\in U$. A vertex $v \in V(G)$ is s-extreme if $V(G)-v$ is a stress convex set in $G$. The stress number $sn(G)$ of $G$ is the minimum cardinality of a set $U$ where $\bigcup_{u,v \in U}S(u,v)=V(G)$. The stress hull number $sh(G)$ of $G$ is the minimum cardinality of a set whose stress convex hull is $V(G)$. In this paper, we present many basic properties of stress intervals. We characterize s-extreme vertices of a graph $G$ and construct graphs $G$ with arbitrarily large difference between the number of s-extreme vertices, $sh(G)$ and $sn(G)$. Then we study these three invariants for some special graph families, such as graph products, split graphs, and block graphs. We show that in any split graph $G$, $sh(G)=sn(G)=|Ext_s(G)|$, where $Ext_s(G)$ is the set of s-extreme vertices of $G$. Finally, we show that for $k \in \mathbb{N}$, deciding whether $sn(G) \leq k$ is NP-complete problem, even when restricted to bipartite graphs.

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arXiv2024

Graphs with span 1 and shortest optimal walks

A span of a given graph $G$ is the maximum distance that two players can keep at all times while visiting all vertices (edges) of $G$ and moving according to certain rules, that produce different variants of span. We prove that the vertex and edge span of the same variant can differ by at most 1 and present a graph where the difference is exactly 1. For all variants of vertex span we present a lower bound in terms of the girth of the graph. Then we study graphs with the strong vertex span equal to 1. We present some nice properties of such graphs and show that interval graphs are contained in the class of graphs having the strong vertex span equal to 1. Finally, we present an algorithm that returns the minimum number of moves needed such that both players traverse all vertices of the given graph $G$ such that in each move the distance between players equals at least the chosen span of $G$.

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arXiv2024

Isolation game on graphs

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 $ι_{\rm g}(G,\cal F)$. 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 $ι_{\rm g}(G,{\cal{F}}') \leq ι_{\rm g}(G,{\cal{F}})$ for any graph $G$. A special focus is given on the isolation game, which takes place when ${\cal F}=\{K_2\}$. We prove that $ι_{\rm g}(G,\{K_2\})\le |V(G)|/2$ 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 $ι_{\rm g}(P_n,\{K_2\})=\left\lfloor\frac{2n+2}{5}\right\rfloor$ when $n \equiv i \pmod 5$ and $i \in \{1,2,3\}$.

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arXiv2024

Induced matching vs edge open packing: trees and product graphs

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, $ρ_{e}^{o}(G)$, 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, $ν_I(G)$, of $G$. While the inequality $ρ_e^o(G)\geq ν_{I}(G)$ 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 $α(G)ν_I(H)$. 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 $ν_I(G\times H)$ and $ρ_{e}^{o}(G\times H)$, 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 $ρ_e^o(Q_n)$ 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.

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arXiv2023

$k$-Domination invariants on Kneser graphs

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, ${γ_{k}}(K(n,r))$, and the $k$-tuple total domination number, ${γ_{t\times k}}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that ${γ_{k}}(K(n,r))\ge {γ_{k}}(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and ${γ_{t\times k}}(K(n,r))\ge {γ_{t\times k}}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that ${γ_{k}}(K(n,r))={γ_{t\times k}}(K(n,r))=k+r$ when $n\geq r(k+r)$, and that in this case every ${γ_{k}}$-set and ${γ_{t\times k}}$-set is a clique, while ${γ_{k}}(r(k+r)-1,r)={γ_{t\times k}}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the 2-packing number, $ρ_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $ρ_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $ρ_2(K(n,r))$ 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.

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arXiv2023

Orientable total domination in graphs

Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $γ_t(D)$, is the minimum cardinality among all total dominating sets of $D$. Given an undirected graph $G$, we study the maximum and minimum total domination numbers among all orientations of $G$. That is, we study the upper (or lower) orientable domination number of $G$, $\rm{DOM}_t(G)$ (or $\rm{dom}_t(G)$), which is the largest (or smallest) total domination number over all orientations of $G$. We characterize those graphs with $\rm{DOM}_t(G) =\rm{dom}_t(G)$ when the girth is at least $7$ as well as those graphs with $\rm{dom}_t(G) = |V(G)|-1$. We also consider how these parameters are effected by removing a vertex from $G$, give exact values of $\rm{DOM}_t(K_{m,n})$ and $\rm{dom}_t(K_{m,n})$ and bound these parameters when $G$ is a grid graph.

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arXiv2023

Bounds on zero forcing using (upper) total domination and minimum degree

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 $γ_t(G)$ (resp. $Γ_t(G)$) of $G$. We prove that $Z(G)+γ_t(G)\le n(G)$ and $Z(G)+\frac{Γ_t(G)}{2}\le n(G)$ 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 $Z(G)+\frac{Γ_t(G)}{2}=n(G)$. 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 $Z(G)\ge δ(G)$. 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.

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