arXiv subjects

Sandi Klavžar

Publications and source records attributed to Sandi Klavžar.

At least 91 records · Page 5Linked to original sources

arXiv2023

A characterization of 4-$χ_S$-vertex-critical graphs for packing sequences with $s_1 =1$ and $s_2\ge 3$

If $S=(s_1,s_2,\ldots)$ is a non-decreasing sequence of positive integers, then the $S$-packing $k$-coloring of a graph $G$ is a mapping $c: V(G)\rightarrow[k]$ such that if $c(u)=c(v)=i$ for $u\neq v\in V(G)$, then $d_G(u,v)>s_i$. The $S$-packing chromatic number of $G$ is the smallest integer $k$ such that $G$ admits an $S$-packing $k$-coloring. A graph $G$ is $χ_S$-vertex-critical if $χ_S(G-u) < χ_S(G)$ for each $u\in V(G)$. If $G$ is $χ_S$-vertex-critical and $χ_S(G) = k$, then $G$ is $k$-$χ_S$-vertex-critical. In this paper, $4$-$χ_S$-vertex-critical graphs are characterized for sequences $S = (1,s_2, s_3, \ldots)$ with $s_2 \ge 3$. There are $28$ sporadic examples and two infinite families of such graphs.

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arXiv2023

Edge general position sets in Fibonacci and Lucas cubes

A set of edges $X\subseteq E(G)$ of a graph $G$ is an edge general position set if no three edges from $X$ lie on a common shortest path in $G$. The cardinality of a largest edge general position set of $G$ is the edge general position number of $G$. In this paper edge general position sets are investigated in partial cubes. In particular it is proved that the union of two largest $Θ$-classes of a Fibonacci cube or a Lucas cube is a maximal edge general position set.

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arXiv2023

The Cut Method on Hypergraphs for the Wiener Index

The cut method has been proved to be extremely useful in chemical graph theory. In this paper the cut method is extended to hypergraphs. More precisely, the method is developed for the Wiener index of $k$-uniform partial cube-hypergraphs. The method is applied to cube-hypergraphs and hypertrees. Extensions of the method to hypergraphs arising in chemistry which are not necessary $k$-uniform and/or not necessary linear are also developed.

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arXiv2023

Computational complexity aspects of super domination

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super dominating set of $G$ is the super domination number of $G$. An exact formula for the super domination number of a tree $T$ is obtained and demonstrated that a smallest super dominating set of $T$ can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph $G$ is at most a given integer if $G$ is a bipartite graph of girth at least $8$. The super domination number is determined for all $k$-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determine.

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arXiv2022

Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products

If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $μ_{\rm t}(G)$ of $G$. Graphs with $μ_{\rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $μ_{\rm t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $μ_{\rm t}(T\,\square\, H) = μ_{\rm t}(T)μ_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $μ_{\rm t}(G\,\square\, H)$ can be arbitrary larger than $μ_{\rm t}(G)μ_{\rm t}(H)$.

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arXiv2022

On mixed metric dimension in subdivision, middle, and total graphs

Let $G$ be a graph and let $S(G)$, $M(G)$, and $T(G)$ be the subdivision, the middle, and the total graph of $G$, respectively. Let ${\rm dim}(G)$, ${\rm edim}(G)$, and ${\rm mdim}(G)$ be the metric dimension, the edge metric dimension, and the mixed metric dimension of $G$, respectively. In this paper, for the subdivision graph it is proved that $\frac{1}{2}\max\{{\rm dim}(G),{\rm edim}(G)\}\leq{\rm mdim}(S(G))\leq{\rm mdim}(G)$. A family of graphs $G_n$ is constructed for which ${\rm mdim}(G_n)-{\rm mdim}(S(G_n))\ge 2$ holds and this shows that the inequality ${\rm mdim}(S(G))\leq{\rm mdim}(G)$ can be strict, while for a cactus graph $G$, ${\rm mdim}(S(G))={\rm mdim}(G)$. For the middle graph it is proved that ${\rm dim}(M(G))\leq{\rm mdim}(G)$ holds, and if $G$ is tree with $n_1(G)$ leaves, then ${\rm dim}(M(G))={\rm mdim}(G)=n_1(G)$. Moreover, for the total graph it is proved that ${\rm mdim}(T(G))=2n_1(G)$ and ${\rm dim}(G)\leq{\rm dim}(T(G))\leq n_1(G)$ hold when $G$ is a tree.

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arXiv2022

Nonlocal metric dimension of graphs

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with ${\rm dim}_{\rm n\ell}(G) = 1$ or with ${\rm dim}_{\rm n\ell}(G) = n(G)-2$ are characterized. The nonlocal metric dimension is determined for block graphs, for corona products, and for wheels. Two upper bounds on the nonlocal metric dimension are proved. An embedding of an arbitrary graph into a supergraph with a small nonlocal metric dimension and small diameter is presented.

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arXiv2022

New transmission irregular chemical graphs

The transmission of a vertex $v$ of a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is a transmission irregular graph. It is shown that for any odd number $n\geq 7$ there exists a transmission irregular chemical tree of order $n$. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge.

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arXiv2022

Orientable domination in product-like graphs

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].

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arXiv2022

Thresholds for the monochromatic clique transversal game

We study a recently introduced two-person combinatorial game, the $(a,b)$-monochromatic clique transversal game which is played by Alice and Bob on a graph $G$. As we observe, this game is equivalent to the $(b,a)$-biased Maker-Breaker game played on the clique-hypergraph of $G$. Our main results concern the threshold bias $a_1(G)$ that is the smallest integer $a$ such that Alice can win in the $(a,1)$-monochromatic clique transversal game on $G$ if she is the first to play. Among other results, we determine the possible values of $a_1(G)$ for the disjoint union of graphs, prove a formula for $a_1(G)$ if $G$ is triangle-free, and obtain the exact values of $a_1(C_n \,\square\, C_m)$, $a_1(C_n \,\square\, P_m)$, and $a_1(P_n \,\square\, P_m)$ for all possible pairs $(n,m)$.

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arXiv2022

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

The total domination number $γ_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that there exist $k$ different vertices in $G$ with the degree sum at most $m(G)$. It is conjectured that $γ_{t}(G)\leq a(G)+1$ holds for every nontrivial connected graph $G$. The conjecture has been proved for graphs with minimum degree at least $3$, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.

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arXiv2022

Extremal results on stepwise transmission irregular graphs

The transmission ${\rm Tr}_G(v)$ of a vertex $v$ of a connected graph $G$ is the sum of distances between $v$ and all other vertices in $G$. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)| =1$ holds for each edge $uv \in E(G)$. In this paper, extremal results on STI graphs with respect to the size and different metric properties are proved. Two extremal families appear in all the cases, balanced complete bipartite graphs of odd order and the so called odd hatted cycles.

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arXiv2021

Twin Vertices in Fault-Tolerant Metric Sets and Fault-Tolerant Metric Dimension of Multistage Interconnection Networks

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 $β'(G) = n(G)$ from [Mathematics 7(1) (2019) 78]. This FTMD problem is reinvestigated for Butterfly networks, Benes networks, and silicate networks. This extends partial results from [IEEE Access 8 (2020) 145435--145445], and at the same time, disproves related conjectures from the same paper.

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arXiv2021

Predominating a vertex in the connected domination game

The connected domination game is played just as the domination game, with an additional requirement that at each stage of the game the vertices played induce a connected subgraph. The number of moves in a D-game (an S-game, resp.) on a graph $G$ when both players play optimally is denoted by $γ_{\rm cg}(G)$ ($γ_{\rm cg}'(G)$, resp.). Connected Game Continuation Principle is established as a substitute for the classical Continuation Principle which does not hold for the connected domination game. Let $G|x$ denote the graph $G$ together with a declaration that the vertex $x$ is already dominated. The first main result asserts that if $G$ is a graph with $γ_{\rm cg}(G) \geq 3$ and $x \in V(G)$, then $γ_{\rm cg}(G|x) \leq 2 γ_{\rm cg}(G) - 3$ and the bound is sharp. The second main theorem states that if $G$ is a graph with $n(G) \geq 2$ and $x \in V(G)$, then $γ_{\rm cg}(G|x) \geq \left \lceil \frac12 γ_{\rm cg}(G) \right \rceil$ and the bound is sharp. Graphs $G$ and their vertices $x$ for which $γ_{\rm cg}'(G|x) = \infty$ holds are also characterized.

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arXiv2021

Zero forcing number versus general position number in tree-like graphs

Let ${\rm Z}(G)$ and ${\rm gp}(G)$ be the zero forcing number and the general position number of a graph $G$, respectively. Known results imply that ${\rm gp}(T)\ge {\rm Z}(T) + 1$ holds for every nontrivial tree $T$. It is proved that the result extends to block graphs. For connected, unicyclic graphs $G$ it is proved that ${\rm gp}(G) \ge {\rm Z}(G)$. The result extends neither to bicyclic graphs nor to quasi-trees. Nevertheless, a large class of quasi-trees is found for which ${\rm gp}(G) \ge {\rm Z}(G)$ holds.

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arXiv2021

On the Chromatic Vertex Stability Number of Graphs

The chromatic vertex (resp.\ edge) stability number ${\rm vs}_χ(G)$ (resp.\ ${\rm es}_χ(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $χ(H)=χ(G)-1$. In the main result it is proved that if $G$ is a graph with $χ(G) \in \{ Δ(G), Δ(G)+1 \}$, then ${\rm vs}_χ(G) = {\rm ivs}_χ(G)$, where ${\rm ivs}_χ(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $χ(G) \le \frac{Δ(G)+1}{2}$. It is proved that if $χ(G) > \frac{Δ(G)}{2}+1$, then ${\rm vs}_χ(G) = {\rm es}_χ(G)$. A Nordhaus-Gaddum-type result on the chromatic vertex stability number is also given.

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arXiv2021

On the computational complexity of the Steiner $k$-eccentricity

The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V (G)$ which contain $v$. A linear time algorithm for calculating the Steiner $k$-eccentricity of a vertex on block graphs is presented. For general graphs, an $O(n^{ν(G)+1}(n(G) + m(G) + k))$ algorithm is designed, where $ν(G)$ is the cyclomatic number of $G$. A linear algorithm for computing the Steiner $3$-eccentricities of all vertices of a tree is also presented which improves the quadratic algorithm from [Discrete Appl.\ Math.\ 304 (2021) 181--195].

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arXiv2021

The general position achievement game played on graphs

A general position set of a graph $G$ is a set of vertices $S$ in $G$ such that no three vertices from $S$ lie on a common shortest path. In this paper we introduce and study the general position achievement game. The game is played on a graph $G$ by players A and B who alternatively pick vertices of $G$. A selection of a vertex is legal if has not been selected before and the set of vertices selected so far forms a general position set of $G$. The player who selects the last vertex wins the game. Playable vertices at each step of the game are described, and sufficient conditions for each of the players to win is given. The game is studied on Cartesian and lexicographic products. Among other results it is proved that A wins the game on $K_n\,\square\, K_m$ if and only if both $n$ and $m$ are odd, and that B wins the game on $G\circ K_n$ if and only if either B wins on $G$ or $n$ is even.

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