arXiv subjects

Dragana Božović

Publications and source records attributed to Dragana Božović.

6 recordsLinked to original sources

arXiv2026

On $k$-limited domination: complexity and Sierpiński graphs

A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by $γ_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set problem is $\mathsf{NP}$-complete, answering an open question posed in the literature. We further show that, for every positive integer $k$, the $k$-Limited Dominating Set problem is $\mathsf{NP}$-complete even when restricted to planar graphs. In addition, we study the $k$-limited domination number of Sierpiński graphs. We determine the exact value of $γ_1^L(S(n,m))$ for all integers $n\ge 1$ and $m\ge 2$, and obtain the $k$-limited domination number for the planar family $S(n,3)$.

math.CO↗

arXiv2026

On externally supported independence number of graphs

We introduce the \emph{externally supported independence number} $α_{\rm es}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $ι(G)$. We show that computing $α_{\rm es}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $α_{\rm es}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely describe the extreme graph classes attaining these bounds.

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arXiv2026

On $k$-limited domination in graphs

In this work, we introduce and study the notion of $k$-limited domination in graphs, motivated by applications where dominating vertices have bounded capacity and cannot be overloaded by too many external neighbors. Formally, given an integer $k \le Δ(G)$, a set of vertices $D \subseteq V(G)$ is called a $k$-limited dominating set if it is a dominating set and, in addition, each vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality of such a set is the $k$-limited domination number, denoted by $γ_k^{\mathrm{L}}(G)$. Since $Δ(G)$-limited domination coincides with the classical domination, we restrict our attention to the nontrivial range $1 \le k < Δ(G)$, where the degree limitation becomes meaningful and leads to new combinatorial phenomena. In this paper, we initiate the study of this concept by deriving sharp general bounds for $γ_k^{\mathrm{L}}(G)(G)$ and identifying conditions under which these bounds can be further improved. We establish a connection between $k$-limited domination and $(1,t)$-domination. In particular, for $d$-regular graphs we prove that $γ_k^{\mathrm{L}}(G)(G)=γ_{1,d-k}(G)$. In the special case $k=1$, we show that $1$-limited domination is tightly linked to graph packings, yielding the bound $γ^{\mathrm L}_1(G) \le n - ρ(G)$ and its characterization. The study reveals several natural open questions and indicates that limited domination provides a rich ground for further research.

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arXiv2025

New results on the Wiener index of trees with a given diameter

We study the Wiener index of a class of trees with fixed diameter and order. A double broom is a tree such that there exist two vertices $u$ and $v$, such that each leaf of $T$ is adjacent to $u$ or $v$. We prove that for a tree $T$ of diameter $d$ and (sufficiently large) order $n$ such that $n\leq d-2+4 \left\lfloor \sqrt{ \frac{d-1}{2}} \right\rfloor$, $T$ has maximum Wiener index (in the class of trees of diameter $d$ and order $n$) if and only if $T$ is a balanced double broom. Our results are sharp up to a small constant.

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arXiv2025

On the star b-chromatic number of a graph

A star coloring of a graph $G$ is a proper coloring where vertices of every two color classes induce a forest of stars. A strict partial order is defined on the set of all star colorings of $G$. We introduce the star b-chromatic number $S_b(G)$, analogous to the b-chromatic number, as the maximum number of colors in a minimum element of the mentioned order. We present several combinatorial properties of $S_b(G)$, compute the exact value for $S_b(G)$ for several known families and compare $S_b(G)$ with several invariants naturally connected to $S_b(G)$.

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arXiv2019

A note on the packing chromatic number of lexicographic products

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. In this short note we present upper and lower bound for the packing chromatic number of the lexicographic product $G\circ H$ of graphs $G$ and $H$. Both bounds coincide in many cases. In particular this happens if $|V(H)|-α(H)\geq {\rm diam}(G)-1$, where $α(G)$ denotes the independence number of $G$.

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