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Sandi Klavžar

Publications and source records attributed to Sandi Klavžar.

At least 109 records · Page 6Linked to original sources
arXiv2021

Global forcing number for maximal matchings in corona products

A global forcing set for maximal matchings of a graph $G=(V(G), E(G))$ is a set $S \subseteq E(G)$ such that $M_1\cap S \neq M_2 \cap S$ for each pair of maximal matchings $M_1$ and $M_2$ of $G$. The smallest such set is called a minimum global forcing set, its size being the global forcing number for maximal matchings $ϕ_{gm}(G)$ of $G$. In this paper, we establish lower and upper bounds on the forcing number for maximal matchings of the corona product of graphs. We also introduce an integer linear programming model for computing the forcing number for maximal matchings of graphs.

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arXiv2021

On the average Steiner 3-eccentricity of trees

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 $v$. In this paper Steiner $3$-eccentricity is studied on trees. Some general properties of the Steiner $3$-eccentricity of trees are given. A tree transformation which does not increase the average Steiner $3$-eccentricity is given. As its application, several lower and upper bounds for the average Steiner $3$-eccentricity of trees are derived.

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arXiv2021

A Steiner general position problem in graph theory

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 $k$-Steiner general position number is determined for trees, cycles and joins of graphs. Lower bounds are presented for split graphs, infinite grids and lexicographic products. The lower bound for the latter products leads to an exact formula for the general position number of an arbitrary lexicographic product.

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arXiv2021

Edge general position problem

Given a graph $G$, the general position problem is to find a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is called a ${\rm gp}$-$set$ of $G$ and its cardinality is the ${\rm gp}$-$number$, ${\rm gp}(G)$, of $G$. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem. The edge general position number, ${\rm gp_{e}}(G)$, is the size of a largest edge general position set of $G$. It is proved that ${\rm gp_{e}}(Q_r) = 2^r$ and that if $T$ is a tree, then ${\rm gp_{e}}(T)$ is the number of its leaves. The value of ${\rm gp_{e}}(P_r\, \square\, P_s)$ is determined for every $r,s\ge 2$. To derive these results, the theory of partial cubes is used. Mulder's meta-conjecture on median graphs is also discussed along the way.

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arXiv2021

On general position sets in Cartesian products

The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r,s\ge 2$, is also determined.

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arXiv2021

On the Relation Between Wiener Index and Eccentricity of a Graph

The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type result on $W(G)$ involving $\varepsilon(G)$ is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $W(T) - \varepsilon(T)$ is minimized on caterpillars. An exact formula for $W(T) - \varepsilon(T)$ in terms of the radius of a tree $T$ is obtained. A lower bound on the eccentricity of a tree in terms of its radius is also given. Two conjectures are proposed. The first asserts that the difference $W(G) - \varepsilon(G)$ does not increase after contracting an edge of $G$. The second conjecture asserts that the difference between the Wiener index of a graph and its eccentricity is largest on paths.

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arXiv2021

Some binary products and integer linear programming for computing $k$-metric dimension of graphs

Let $G$ be a connected graph. For an ordered set $S=\{v_1,\ldots, v_\ell\}\subseteq V(G)$, the vector $r_G(v|S) = (d_G(v_1,v), \ldots, d_G(v_\ell,v))$ is called the metric $S$-representation of $v$. If for any pair of different vertices $u,v\in V(G)$, the vectors $r(v|S)$ and $r(u|S)$ differ in at least $k$ positions, then $S$ is a $k$-metric generator for $G$. A smallest $k$-metric generator for $G$ is a $k$-{\em metric basis} for $G$, its cardinality being the $k$-metric dimension of $G$. A sharp upper bound and a closed formulae for the $k$-metric dimension of the hierarchical product of graphs is proved. Also, sharp lower bounds for the $k$-metric dimension of the splice and link products of graphs are presented. An integer linear programming model for computing the $k$-metric dimension and a $k$-metric basis of a given graph is proposed. These results are applied to bound or to compute the $k$-metric dimension of some classes of graphs that are of interest in mathematical chemistry.

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arXiv2021

The domination game played on diameter 2 graphs

Let $γ_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $γ_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if ${\rm diam}(G) = 2$ and $n(G) \le 10$, then $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ if and only if $G$ is one of seven sporadic graphs with $n(G)\le 6$ or the Petersen graph, and there are exactly ten graphs of diameter $2$ and order $11$ that attain the bound.

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arXiv2021

The geodesic-transversal problem

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.

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arXiv2020

Comparing Wiener complexity with eccentric complexity

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. The Wiener complexity of $G$ is the number of different transmissions of its vertices. Similarly, the eccentric complexity of $G$ is defined as the number of different eccentricities of its vertices. In this paper these two complexities are compared. The complexities are first studied on Cartesian product graphs. Transmission indivisible graphs and arithmetic transmission graphs are introduced to demonstrate sharpness of upper and lower bounds on the Wiener complexity, respectively. It is shown that for almost all graphs the Wiener complexity is not smaller than the eccentric complexity. This property is proved for trees, the equality holding precisely for center-regular trees. Several families of graphs in which the complexities are equal are constructed. Using the Cartesian product, it is proved that the eccentric complexity can be arbitrarily larger than the Wiener complexity. Additional infinite families of graphs with this property are constructed by amalgamating universally diametrical graphs with center-regular trees.

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arXiv2020

$1/2$-conjectures on the domination game and claw-free graphs

Let $γ_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $γ_g(G) \le \left\lceil \frac{1}{2}n(G)\right\rceil$. Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if $δ(G) \geq 2$, then $γ_g(G) \leq \left\lceil \frac{1}{2}n(G) \right\rceil$. We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound $γ_g(G) \le \left\lceil \frac{11}{20} \, n(G) \right\rceil$ over the class of claw-free graphs of minimum degree at least $2$. Computer experiments supporting the new conjecture and sharpness examples are also presented.

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arXiv2020

Total connected domination game

The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($γ_{\rm tcg}(G)$) $γ_{\rm cg}(G)$ of $G$. We show that $γ_{\rm tcg}(G)\in \{γ_{\rm cg}(G), γ_{\rm cg}(G) + 1, γ_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class $i$ if $γ_{\rm tcg}(G) = γ_{\rm cg} + i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class $2$ and characterize Class $1$ trees. We provide an infinite family of Class $2$ bipartite graphs.

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arXiv2020

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed.

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arXiv2020

The general position number of the Cartesian product of two trees

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

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arXiv2020

The Steiner $k$-eccentricity on trees

We study the Steiner $k$-eccentricity on trees, which generalizes the previous one in the paper [X.~Li, G.~Yu, S.~Klavžar, On the average Steiner 3-eccentricity of trees, arXiv:2005.10319, 2020]. To support the algorithm, we achieve much stronger properties for the Steiner $k$-ecc tree than that in the previous paper. Based on this, a linear time algorithm is devised to calculate the Steiner $k$-eccentricity of a vertex in a tree. On the other hand, the lower and upper bounds of the average Steiner $k$-eccentricity index of a tree on order $n$ are established based on a novel technique which is quite different from that in the previous paper but much easier to follow.

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arXiv2020

On the Djoković-Winkler relation and its closure in subdivisions of fullerenes, triangulations, and chordal graphs

It was recently pointed out that certain SiO$_2$ layer structures and SiO$_2$ nanotubes can be described as full subdivisions aka subdivision graphs of partial cubes. A key tool for analyzing distance-based topological indices in molecular graphs is the Djoković-Winkler relation $Θ$ and its transitive closure $Θ^\ast$. In this paper we study the behavior of $Θ$ and $Θ^\ast$ with respect to full subdivisions. We apply our results to describe $Θ^\ast$ in full subdivisions of fullerenes, plane triangulations, and chordal graphs.

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arXiv2020

Some extremal results on the chromatic-stability index

The $χ$-stability index ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J.\ Combin.\ 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($k\le 5$) graphs $G$ with ${\rm es}_χ(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with $χ(G)=3$ for which ${\rm es}_χ(G)+{\rm es}_χ(\overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${\rm es}_χ(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $n\equiv 2 \pmod 3$ and $χ(G)=3$.

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