arXiv subjects

Sandi Klavžar

Publications and source records attributed to Sandi Klavžar.

At least 127 records · Page 7Linked to original sources

arXiv2020

On $\ell$-distance balanced product graphs

A graph $G$ is $\ell$-distance-balanced if for each pair of vertices $x$ and $y$ at distance $\ell$ in $G$, the number of vertices closer to $x$ than to $y$ is equal to the number of vertices closer to $y$ than to $x$. A complete characterization of $\ell$-distance-balanced corona products is given and a characterization of lexicographic products for $\ell \ge 3$, thus complementing known results for $\ell\in \{1,2\}$ and correcting an earlier related assertion. A sufficient condition on $H$ which guarantees that $K_n \,\square\, H$ is $\ell$-distance-balanced is given and it is proved that if $K_n \,\square\, H$ is $\ell$-distance-balanced, then $H$ is an $\ell$-distance-balanced graph. A known characterization of $1$-distance-balanced graphs is extended to $\ell$-distance-balanced graphs, again correcting an earlier claimed assertion.

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arXiv2020

On Rall's $1/2$-conjecture on the domination game

The $1/2$-conjecture on the domination game asserts that if $G$ is a traceable graph, then the game domination number $γ_g(G)$ of $G$ is at most $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a $1/2$-graph if $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ holds. It is proved that the so-called hatted cycles are $1/2$-graphs and that unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of graphs that support the conjecture are determined and computer experiments related to the conjecture described.

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arXiv2020

Maker-Breaker resolving game

A set of vertices $W$ of a graph $G$ is a resolving set if every vertex of $G$ is uniquely determined by its vector of distances to $W$. In this paper, the Maker-Breaker resolving game is introduced. The game is played on a graph $G$ by Resolver and Spoiler who alternately select a vertex of $G$ not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of $G$, whereas Spoiler wins if the Resolver cannot form a resolving set of $G$. The outcome of the game is denoted by $o(G)$ and $R_{\rm MB}(G)$ (resp. $S_{\rm MB}(G)$) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by $R'_{\rm MB}(G)$ and $S'_{\rm MB}(G)$. Invariants $R_{\rm MB}(G)$, $R'_{\rm MB}(G)$, $S_{\rm MB}(G)$, and $S'_{\rm MB}(G)$ are compared among themselves and with the metric dimension ${\rm dim}(G)$. A large class of graphs $G$ is constructed for which $R_{\rm MB}(G) > {\rm dim}(G)$ holds. The effect of twin equivalence classes and pairing resolving sets on the Maker-Breaker resolving game is described. As an application $o(G)$, as well as $R_{\rm MB}(G)$ and $R'_{\rm MB}(G)$ (or $S_{\rm MB}(G)$ and $S'_{\rm MB}(G)$), are determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

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arXiv2020

The general position number of integer lattices

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 pairwise distinct vertices from $S$ lie on a common geodesic. The $n$-dimensional grid graph $\pn$ is the Cartesian product of $n$ copies of the two-way infinite path $P_\infty$. It is proved that if $n\in {\mathbb N}$, then ${\rm gp}({P_\infty^n}) = 2^{2^{n-1}}$. The result was earlier known only for $n\in \{1,2\}$ and partially for $n=3$.

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arXiv2020

On the difference between the eccentric connectivity index and eccentric distance sum of graphs

The eccentric connectivity index of a graph $G$ is $ξ^c(G) = \sum_{v \in V(G)}\varepsilon(v)°(v)$, and the eccentric distance sum is $ξ^d(G) = \sum_{v \in V(G)}\varepsilon(v)D(v)$, where $\varepsilon(v)$ is the eccentricity of $v$, and $D(v)$ the sum of distances between $v$ and the other vertices. A lower and an upper bound on $ξ^d(G) - ξ^c(G)$ is given for an arbitrary graph $G$. Regular graphs with diameter at most $2$ and joins of cocktail-party graphs with complete graphs form the graphs that attain the two equalities, respectively. Sharp lower and upper bounds on $ξ^d(T) - ξ^c(T)$ are given for arbitrary trees. Sharp lower and upper bounds on $ξ^d(G)+ξ^c(G)$ for arbitrary graphs $G$ are also given, and a sharp lower bound on $ξ^d(G)$ for graphs $G$ with a given radius is proved.

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arXiv2020

Constructing new families of transmission irregular graphs

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Additional infinite families among the starlike trees and bi-starlike trees are determined. Transmission irregular unicyclic infinite families are also presented, in particular, the line graph of $T(a,a+1,a+2)$, $a\ge 2$, is transmission irregular if and only if $a$ is even.

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arXiv2020

Edge metric dimensions via hierarchical product and integer linear programming

If $S=\{v_1,\ldots, v_k\}$ is an ordered subset of vertices of a connected graph $G$ and $e$ is an edge of $G$, then the vector $r_G(e|S) = (d_G(v_1,e), \ldots, d_G(v_k,e))$ is the edge metric $S$-representation of $e$. If the vertices of $G$ have pairwise different edge metric $S$-representations, then $S$ is an edge metric generator for $G$. The cardinality of a smallest edge metric generator is the edge metric dimension ${\rm edim}(G)$ of $G$. A general sharp upper bound on the edge metric dimension of hierarchical products $G(U)\sqcap H$ is proved. Exact formula is derived for the case when $|U| = 1$. An integer linear programming model for computing the edge metric dimension is proposed. Several examples are provided which demonstrate how these two methods can be applied to obtain the edge metric dimensions of some applicable graphs.

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arXiv2020

Dominated and dominator colorings over (edge) corona and hierarchical products

Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chromatic number of hierarchical products with two factors determined. An application of dominated colorings in genetic networks is also proposed.

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arXiv2020

$S$-packing chromatic vertex-critical graphs

For a non-decreasing sequence of positive integers $S = (s_1,s_2,\ldots)$, the {\em $S$-packing chromatic number} $χ_S(G)$ of $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $X_i$, $i \in [k]$, where vertices in $X_i$ are pairwise at distance greater than $s_i$. In this paper we introduce $S$-packing chromatic vertex-critical graphs, $χ_{S}$-critical for short, as the graphs in which $χ_{S}(G-u)<χ_{S}(G)$ for every $u\in V(G)$. This extends the earlier concept of the packing chromatic vertex-critical graphs. We show that if $G$ is $χ_{S}$-critical, then the set $\{ χ_{S}(G)-χ_{S}(G-u); \, u\in V(G) \}$ can be almost arbitrary. If $G$ is $χ_{S}$-critical and $χ_{S}(G)=k$ ($k\in \mathbb{N}$), then $G$ is called $k$-$χ_{S}$-critical. We characterize $3$-$χ_{S}$-critical graphs and partially characterize $4$-$χ_{S}$-critical graphs when $s_1>1$. We also deal with $k$-$χ_{S}$-criticality of trees and caterpillars.

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arXiv2020

On the general position number of complementary prisms

The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is the graph formed from the disjoint union of $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between them. It is proved that ${\rm gp}(G\overline{G})\le n(G) + 1$ if $G$ is connected and ${\rm gp}(G\overline{G})\le n(G)$ if $G$ is disconnected. Graphs $G$ for which ${\rm gp}(G\overline{G}) = n(G) + 1$ holds, provided that both $G$ and $\overline{G}$ are connected, are characterized. A sharp lower bound on ${\rm gp}(G\overline{G})$ is proved. If $G$ is a connected bipartite graph or a split graph then ${\rm gp}(G\overline{G})\in \{n(G), n(G)+1\}$. Connected bipartite graphs and block graphs for which ${\rm gp}(G\overline{G})=n(G)+1$ holds are characterized. A family of block graphs is constructed in which the ${\rm gp}$-number of their complementary prisms is arbitrary smaller than their order.

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arXiv2019

Comparison of Wiener index and Zagreb eccentricity indices

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T) \ge E_2(T)$ holds, and if the diameter of $T$ is large, then $W(T) < E_1(T)$ holds.

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arXiv2019

Z-domination game

The Z-domination game is a variant of the domination game in which each newly selected vertex $u$ in the game must have a not yet dominated neighbor, but after the move all vertices from the closed neighborhood of $u$ are declared to be dominated. The Z-domination game is the fastest among the five natural domination games. The corresponding game Z-domination number of a graph $G$ is denoted by $γ_{Zg}(G)$. It is proved that the game domination number and the game total domination number of a graph can be expressed as the game Z-domination number of appropriate lexicographic products. Graphs with a Z-insensitive property are introduced and it is proved that if $G$ is Z-insensitive, then $γ_{Zg}(G)$ is equal to the game domination number of $G$. Weakly claw-free graphs are defined and proved to be Z-insensitive. As a consequence, $γ_{Zg}(P_n)$ is determined, thus sharpening an earlier related approximate result. It is proved that if $γ_{Zg}(G)$ is an even number, then $γ_{Zg}(G)$ is strictly smaller than the game L-domination number. On the other hand, families of graphs are constructed for which all five game domination numbers coincide. Graphs $G$ with $γ_{Zg}(G) = γ(G)$ are also considered and computational results which compare the studied invariants in the class of trees on at most $16$ vertices reported.

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arXiv2019

Perfect graphs for domination games

Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg}$-perfect graphs are precisely $\overline{2P_3}$-free cographs.

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arXiv2019

Critical graphs for the chromatic edge-stability number

The chromatic edge-stability number ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $χ(G')=χ(G)-1$. Edge-stability critical graphs are introduced as the graphs $G$ with the property that ${\rm es}_χ(G-e) < {\rm es}_χ(G)$ holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph with $χ(G)=k$ and ${\rm es}_χ(G)=\ell$, 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 $χ(G)=k$ 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.

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arXiv2019

The annihilation number does not bound the 2-domination number from the above

The $2$-domination number $γ_2(G)$ of a graph $G$ is the minimum cardinality of a set $S\subseteq V(G)$ such that every vertex from $V(G)\setminus S$ is adjacent to at least two vertices in $S$. The annihilation number $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of its edges. It was conjectured that $γ_2(G) \leq a(G) +1$ holds for every connected graph $G$. The conjecture was earlier confirmed, in particular, for graphs of minimum degree $3$, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the $2$-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of Jakovac from [Discrete Appl.\ Math.\ 260 (2019) 178--187].

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arXiv2019

Characterization of general position sets and its applications to cographs and bipartite graphs

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, ${\rm gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then ${\rm gp}(G) \leq α(G)$ with equality if ${\rm diam}(G) \in \{2,3\}$. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.

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arXiv2019

A Lower Bound and Several Exact Results on the $d$-Lucky Number

If $\ell: V(G)\rightarrow {\mathbb N}$ is a vertex labeling of a graph $G = (V(G), E(G))$, then the $d$-lucky sum of a vertex $u\in V(G)$ is $d_\ell(u) = d_G(u) + \sum_{v\in N(u)}\ell(v)$. The labeling $\ell$ is a $d$-lucky labeling if $d_\ell(u)\neq d_\ell(v)$ for every $uv\in E(G)$. The $d$-lucky number $η_{dl}(G)$ of $G$ is the least positive integer $k$ such that $G$ has a $d$-lucky labeling $V(G)\rightarrow [k]$. A general lower bound on the $d$-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinite family of corona graphs. The $d$-lucky number is also determined for the so-called $G_{n,m}$-web graphs and graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.

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arXiv2019

The general position problem on Kneser graphs and on some graph operations

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number (gp-number) ${\rm gp}(G)$ of $G$. The gp-number is determined for some families of Kneser graphs, in particular for $K(n,2)$ and $K(n,3)$. A sharp lower bound on the gp-number is proved for Cartesian products of graphs. The gp-number is also determined for joins of graphs, coronas over graphs, and line graphs of complete graphs.

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