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Sandi Klavžar

Publications and source records attributed to Sandi Klavžar.

At least 145 records · Page 8Linked to original sources
arXiv2019

Maker-Breaker domination number

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $γ_{\rm MB}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $γ_{\rm MB}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $γ_{\rm MB}(G)$ is also compared with the domination number. Using the Erdős-Selfridge Criterion a large class of graphs $G$ is found for which $γ_{\rm MB}(G) > γ(G)$ holds. Residual graphs are introduced and used to bound/determine $γ_{\rm MB}(G)$ and $γ_{\rm MB}'(G)$. Using residual graphs, $γ_{\rm MB}(T)$ and $γ_{\rm MB}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

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arXiv2019

Local metric dimension of graphs: generalized hierarchical products and some applications

Let $G$ be a graph and $S\subseteq V(G)$. If every two adjacent vertices of $G$ have different metric $S$-representations, then $S$ is a local metric generator for $G$. A local metric generator of smallest order is a local metric basis for $G$, its order is the local metric dimension of $G$. Lower and upper bounds on the local metric dimension of the generalized hierarchical product are proved and demonstrated to be sharp. The results are applied to determine or bound the dimension of several graphs of importance in mathematical chemistry. Using the dimension, a new model for assigning codes to customers in delivery services is proposed.

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arXiv2019

Packing chromatic vertex-critical graphs

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $χ_ρ$-critical for short, are introduced as the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. If $χ_ρ(G) = k$, then $G$ is $k$-$χ_ρ$-critical. It is shown that if $G$ is $χ_ρ$-critical, then the set $\{χ_ρ(G) - χ_ρ(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$χ_ρ$-critical graphs are characterized, and $4$-$χ_ρ$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$χ_ρ$-critical tree and that a $k$-$χ_ρ$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le χ_ρ(G)$, then $G\,\square\, H$ is $χ_ρ$-critical.

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arXiv2019

Maker-Breaker total domination game

Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duchêne, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinitely many connected cubic graphs in which Staller wins and that no minimum degree condition is sufficient to guarantee that Dominator wins when Staller starts the game. An amalgamation lemma is established and used to determine the outcome of the game played on grids. Cacti are also classified with respect to the outcome of the game. A connection between the game and hypergraphs is established. It is proved that the game is PSPACE-complete on split and bipartite graphs. Several problems and questions are also posed.

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arXiv2018

Distribution of global defensive $k$-alliances over some graph products

If $G=(V_G, E_G)$ is a graph, then $S\subseteq V_G$ is a global defensive $k$-alliance in $G$ if (i) each vertex not in $S$ has a neighbor in $S$ and (ii) each vertex of $S$ has at least $k$ more neighbors inside $S$ than outside of it. The global defensive $k$-alliance number of $G$ is the minimum cardinality among all global defensive $k$-alliance in $G$. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds are also discussed.

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arXiv2018

Domination game and minimal edge cuts

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if $C$ a minimum edge cut of a connected graph $G$, then $γ_g(G) \le γ_g(G\setminus C) + 2κ'(G)$. Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.

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arXiv2018

Exact distance graphs of product graphs

Given a graph $G$, the exact distance-$p$ graph $G^{[\natural p]}$ has $V(G)$ as its vertex set, and two vertices are adjacent whenever the distance between them in $G$ equals $p$. We present formulas describing the structure of exact distance-$p$ graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-$2$ graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-$p$ graphs of hypercubes $Q_n$ are also studied. As these graphs contain generalized Johnson graphs as induced subgraphs, we use some known and find some new constructions of their colorings. These constructions are applied for colorings of the exact distance-$p$ graphs of hypercubes with the focus on the chromatic number of $Q_{n}^{[\natural p]}$ for $p\in \{n-2,n-3,n-4\}$.

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arXiv2018

M-Polynomial Revisited: Bethe Cacti and an Extension of Gutman's Approach

The $M$-polynomial of a graph $G$ is defined as $\sum_{i\le j} m_{i,j}(G)x^iy^j$, where $m_{i,j}(G)$, $i,j\ge 1$, is the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. Knowing the $M$-polynomial, formulas for bond incident degree indices (an important subclass of degree-based topological indices) can be obtained by means of specific operators defined on differentiable functions in two variables. This is illustrated on three infinite families of Bethe cacti. Gutman's approach for the computation of the coefficients of the $M$-polynomial is also recalled and an extension of it is given. This extension is used to determine the $M$-polynomial of a two-parameter infinite family of lattice graphs.

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arXiv2018

The variety of domination games

Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than $6/7$ of the order of a graph.

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arXiv2018

Strong geodetic cores and Cartesian product graphs

The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that ${\rm sg}(G \,\square\, K_2) \geq {\rm sg}(G)$ holds for different families of graphs, a result conjectured to be true in general. Counterexamples are constructed demonstrating that the conjecture does not hold in general.

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arXiv2018

Cutting Lemma and Union Lemma for the Domination Game

Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are listed.

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arXiv2017

On Grundy total domination number in product graphs

A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $γ_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $γ_{gr}^t(G\times H) \geq γ_{gr}^t(G)γ_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express $γ_{gr}^t(G\circ H)$ in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on $γ_{gr}^t(G \boxtimes H)$ are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.

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arXiv2017

The graph theory general position problem on some interconnection networks

Given a graph $G$, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short) of $G$ and denoted by ${\rm gp}(G)$. In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal grid. To derive these results, we introduce monotone-geodesic labeling and prove a Monotone Geodesic Lemma that is in turn developed using the Erdös-Szekeres theorem on monotone sequences. The gp-number of the 3-dim infinite grid is bounded. Using isometric path covers, the gp-number is also determined for Beneš networks.

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arXiv2017

Game Total Domination Critical Graphs

In the total domination game played on a graph $G$, players Dominator and Staller alternately select vertices of $G$, as long as possible, such that each vertex chosen increases the number of vertices totally dominated. Dominator (Staller) wishes to minimize (maximize) the number of vertices selected. The game total domination number, $γ_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. If a vertex $v$ of $G$ is declared to be already totally dominated, then we denote this graph by $G|v$. In this paper the total domination game critical graphs are introduced as the graphs $G$ for which $γ_{\rm tg}(G|v) < γ_{\rm tg}(G)$ holds for every vertex $v$ in $G$. If $γ_{\rm tg}(G) = k$, then $G$ is called $k$-$γ_{\rm tg}$-critical. It is proved that the cycle $C_n$ is $γ_{\rm tg}$-critical if and only if $n\pmod 6 \in \{0,1,3\}$ and that the path $P_n$ is $γ_{\rm tg}$-critical if and only if $n\pmod 6\in \{2,4\}$. $2$-$γ_{\rm tg}$-critical and $3$-$γ_{\rm tg}$-critical graphs are also characterized as well as $3$-$γ_{\rm tg}$-critical joins of graphs.

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arXiv2017

Embeddings into almost self-centered graphs of given radius

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius $r$ is called an $r$-ASC graph. The $r$-ASC index $θ_r(G)$ of a graph $G$ is the minimum number of vertices needed to be added to $G$ such that an $r$-ASC graph is obtained that contains $G$ as an induced subgraph. It is proved that $θ_r(G)\le 2r$ holds for any graph $G$ and any $r\ge 2$ which improves the earlier known bound $θ_r(G)\le 2r+1$. It is further proved that $θ_r(G)\le 2r-1$ holds if $r\geq 3$ and $G$ is of order at least $2$. The $3$-ASC index of complete graphs is determined. It is proved that $θ_3(G)\in \{3,4\}$ if $G$ has diameter $2$ and for several classes of graphs of diameter $2$ the exact value of the $3$-ASC index is obtained. For instance, if a graph $G$ of diameter $2$ does not contain a diametrical triple, then $θ_3(G) = 4$. The $3$-ASC index of paths of order $n\geq 1$, cycles of order $n\geq 3$, and trees of order $n\geq 10$ and diameter $n-2$ are also determined, respectively, and several open problems proposed.

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arXiv2017

Graph theory general position problem

The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset Selection Problem is to find a maximum subset $S'$ of $S$ such that no three points of $S'$ are collinear. Motivated by these problems, the following graph theory variation is introduced: Given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a gp-set of $G$ and its size is the gp-number ${\rm gp}(G)$ of $G$. Upper bounds on ${\rm gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

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arXiv2017

Strong geodetic problem in grid like architectures

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If $G$ is a graph, then ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that one can assign a fixed geodesic to each pair $\{x,y\}\subseteq S$ so that these $\binom{|S|}{2}$ geodesics cover all the vertices of $G$. In this paper, the strong geodesic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on flat grids and flat cylinders.

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