arXiv subjects

Gordana Radić

Publications and source records attributed to Gordana Radić.

3 recordsLinked to original sources

arXiv2026

An improved lower and upper bound of the k-limited domination number

We continue the study of $k$-limited domination in graphs, a domination variant in which each vertex of a dominating set may dominate at most $k$ vertices outside the set. This concept extends classical domination by incorporating capacity constraints on dominating vertices. We improve the general lower bound for the $k$-limited domination number $γ_k^L(G)$ by employing the concept of $k$-capacitated domination. As a consequence, we characterize graphs satisfying $γ_k^L(G)=\lceil \frac{n}{k+1} \rceil$. We also refine the known upper bound for graphs with $k<δ(G)$ using the $k$-limited packing number. Under this condition, we describe graphs attaining $γ_k^L(G)=n-k$. These results extend and unify previous investigations for the case $k=1$ and provide a complete characterization of graphs attaining the extreme values of the $k$-limited domination number under the considered assumptions.

math.CO↗

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.

math.CO↗

arXiv2025

Maps preserving the idempotency of Jordan products

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X of dimension at least three. For an arbitrary nonzero complex number t we determine the form of mappings f: B(X)-->B(X) with sufficiently large range such that t(AB+BA) is idempotent if and only if t(f(A)f(B)+f(B)f(A)) is idempotent, for all A, B in B(X). Note that f is not assumed to be linear or additive.

math.FA↗