arXiv subjects

Joanna Raczek

Publications and source records attributed to Joanna Raczek.

3 recordsLinked to original sources

arXiv2026

A Graph Theoretic Approach to Spatial Modeling of Post Disaster Shelter Camps Using Rainbow and Roman Domination Parameters

Effective spatial organization of post-disaster shelter camps is essential for ensuring access to basic services while making efficient use of limited space and resources. In this paper, we propose a graph-theoretic framework for shelter-camp facility placement based on rainbow $k$-domination and Roman domination. Rainbow $k$-domination models the simultaneous accessibility of distinct facility types, such as sanitation units, kitchens, water points, and schools, whereas Roman domination is used to represent services with different capacity levels, illustrated through Wi-Fi deployment. We present an $O(nk)$-time algorithm for finding a minimum rainbow $k$-dominating set of a tree with $n$ vertices, which is linear in $n$ for fixed $k$, and computational experiments confirm its scalability on large instances. For general graphs, we establish new lower and upper bounds on the rainbow $k$-domination number. We further investigate its relationship with Roman domination, derive structural properties of graphs attaining the extremal equality between the two parameters, and prove that recognizing such graphs is NP-hard. These results provide a theoretical foundation for domination-based approaches to facility placement in post-disaster shelter planning.

cs.DM↗

arXiv2019

Paired Domination versus Domination and Packing Number in Graphs

Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $γ(G)$, $γ_{\rm pr}(G)$, and $γ_{t}(G)$, respectively. For a positive integer $k$, a $k$-packing in $G$ is a set $S \subseteq V(G)$ such that for every pair of distinct vertices $u$ and $v$ in $S$, the distance between $u$ and $v$ is at least $k+1$. The $k$-packing number is the order of a largest $k$-packing and is denoted by $ρ_{k}(G)$. It is well known that $γ_{\rm pr}(G) \le 2γ(G)$. In this paper, we prove that it is NP-hard to determine whether $γ_{\rm pr}(G) = 2γ(G)$ even for bipartite graphs. We provide a simple characterization of trees with $γ_{\rm pr}(G) = 2γ(G)$, implying a polynomial-time recognition algorithm. We also prove that even for a bipartite graph, it is NP-hard to determine whether $γ_{\rm pr}(G)=γ_{t}(G)$. We finally prove that it is both NP-hard to determine whether $γ_{\rm pr}(G)=2ρ_{4}(G)$ and whether $γ_{\rm pr}(G)=2ρ_{3}(G)$.

cs.DM↗

arXiv2013

Domination subdivision and domination multisubdivision numbers of graphs

The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown \cite{vel} that sd$(T)\leq 3$ for any tree $T$. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the \emph{domination multisubdivision number} of a nonempty graph $G$ as a minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. We show that msd$(G)\leq 3$ for any graph $G$. The domination subdivision number and the domination multisubdivision numer of a graph are incomparable in general case, but we show that for trees these two parameters are equal. We also determine domination multisubdivision number for some classes of graphs.

math.CO↗