arXiv subjects

Didem Gözüpek

Publications and source records attributed to Didem Gözüpek.

At least 19 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↗

arXiv2026

On the Packing Coloring Gap of Graphs

The packing chromatic number of a graph is the minimum number of colors for which the graph admits a packing coloring. This distance-based parameter may change under local structural modifications of the graph. In this paper, we introduce the packing coloring gap, defined as the maximum decrease in the packing chromatic number caused by the deletion of a single vertex. We focus on trees and determine the packing coloring gap for caterpillars. We further extend these results to caterpillars under the corona operation with K1. In addition, we present examples of graphs with packing coloring gap zero, one, and arbitrarily large.

math.CO↗

arXiv2026

Packing chromatic critical graphs with radius at most 2

For a graph $G$ with vertex set $V(G)$ and a positive integer $i$, an $i$-packing in $G$ is a subset $X$ of $V(G)$ such that the distance between any two distinct vertices of $X$ is greater than $i$. The packing chromatic number of $G$, denoted by $χ_ρ(G)$, is the smallest positive integer $k$ for which there exists a partition $X_1, X_2, \ldots, X_k$ of $V(G)$ such that $X_i$ is an $i$-packing in $G$ for every $i \in [k]$. A graph $G$ is called $χ_ρ$-critical if $χ_ρ(H) < χ_ρ(G)$ holds for every proper subgraph $H$ of $G$. In this paper, we provide a structural characterization of $χ_ρ$-critical graphs with radius $1$, and completely determine the $χ_ρ$-critical cactus graphs with radius $2$ and diameter $2$ or $3$.

math.CO↗

arXiv2026

S-packing chromatic critical paths and cycles

Let $S=(s_1,s_2,\ldots)$ be a non-decreasing sequence of positive integers. For a graph $G$ with vertex set $V(G)$, a labeling $ϕ\colon V(G)\to \{1,\ldots,k\}$ is an $S$-packing $k$-coloring if, whenever two distinct vertices $u,v\in V(G)$ are assigned the same color $i$, their distance in $G$ is greater than $s_i$. The minimum $k$ for which $G$ admits such a coloring is the $S$-packing chromatic number of $G$. A graph $G$ is $χ_S$-vertex-critical if $χ_S(G-v) < χ_S(G)$ for every $v \in V(G)$, and it is $χ_S$-critical if $χ_S(H) < χ_S(G)$ holds for every proper subgraph $H$ of $G$. In this paper, the exact value of $χ_S(P_n)$ is determined for every path of order $n$ and for every packing sequence $S$ where $s_i < 2^i$ holds for each entry $s_i$. As a consequence, $χ_S$-critical and $χ_S$-vertex-critical paths are identified for each such sequence $S$. In addition, we extend earlier results on $χ_S$-critical cycles and provide a complete characterization of $χ_S$-critical and $χ_S$-vertex-critical cycles for packing sequences $S= (1, s_2, \dots )$ with $s_2 \in \{2,3\}$ and $s_3,s_4 \in \{4,5,6,7\}$.

math.CO↗

arXiv2025

Packing coloring of graphs with long paths

The packing coloring problem has diverse applications, including frequency assignment in wireless networks, resource distribution and facility location in smart cities and post-disaster management, as well as in biological diversity. Formally, the packing coloring of a graph is a vertex coloring in which any two vertices assigned color $i$ are at a distance of at least $i+1$, and the smallest number of colors admitting such a coloring is called the packing chromatic number. Goddard et al.~\cite{goddard2008broadcast} showed that the packing chromatic numbers of paths and cycles are at most 3 and 4, respectively. In this paper, we introduce \emph{path-aligned graph products}, a natural extension of paths with unbounded diameter. We extend the result of~\cite{goddard2008broadcast} by proving that the packing chromatic number remains bounded by a constant for several families of path-aligned cycle and path-aligned complete products. We then investigate the packing chromatic number of caterpillars, another class of graphs characterized by long induced paths. Sloper~\cite{sloper} proved that the packing chromatic number of caterpillars is at most 7; here, we provide a complete structural characterization of caterpillars with packing chromatic number at most 3. Finally, several open research questions are posed.

math.CO↗

arXiv2025

S-packing chromatic critical graphs

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $χ_S(G)$. In this paper, $χ_S$-critical graphs are introduced as the graphs $G$ such that $χ_S(H) < χ_S(G)$ for each proper subgraph $H$ of $G$. Several families of $χ_S$-critical graphs are constructed, and $2$- and $3$-colorable $χ_S$-critical graphs are presented for all packing sequences $S$, while $4$-colorable $χ_S$-critical graphs are found for most of $S$. Cycles which are $χ_S$-critical are characterized under different conditions. It is proved that for any graph $G$ and any edge $e \in E(G)$, the inequality $χ_S(G - e) \ge χ_S(G)/2$ holds. Moreover, in several important cases, this bound can be improved to $χ_S(G - e) \ge (χ_S(G)+1)/2$. The sharpness of the bounds is also discussed. Along the way an earlier result on $χ_S$-vertex-critical graphs is supplemented.

math.CO↗

arXiv2025

Grundy Packing Coloring of Graphs

A map $c:V(G)\rightarrow\{1,\dots,k\}$ of a graph $G$ is a packing $k$-coloring if every two different vertices of the same color $i\in \{1,\dots,k\}$ are at distance more than $i$. The packing chromatic number $χ_ρ(G)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of \textit{Grundy packing chromatic number}, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of $G$. We then define the Grundy packing chromatic number $Γ_ρ(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of $Γ_ρ(G)$, provide results on the complexity of the problem as well as bounds and some exact results for $Γ_ρ(G)$.

math.CO↗

arXiv2021

Upper paired domination versus upper domination

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $Γ(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $Γ_{pr}(G)$. By Henning and Pradhan (2019), we know that $Γ_{pr}(G)\leq 2Γ(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $Γ_{pr}(G)= 2Γ(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $Γ_{pr}(G)= 2Γ(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $Γ_{pr}(G)= 2Γ(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $Γ_{pr}(G)= 2Γ(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $Γ_{pr}(G)= 2Γ(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $Γ_{pr}(G)= 2Γ(G)$ as an open question.

math.CO↗

arXiv2021

Minimum Reload Cost Cycle Cover in Complete Graphs

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al.[1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not approximable within $1/ε$ for any $ε> 0$ even when the number of colors is 2, the reload costs are symmetric and satisfy the triangle inequality. In this paper, we study this problem in complete graphs having equitable or nearly equitable 2-edge-colorings, which are edge-colorings with two colors such that for each vertex $v \in V(G)$, $||c_1(v)| -|c_2(v)|| \leq 1$ or $||c_1(v)| -|c_2(v)|| \leq 2$, respectively, where $c_i(v)$ is the set of edges with color $i$ that is incident to $v$. We prove that except possibly on complete graphs with fewer than 13 vertices, the minimum reload cost is zero on complete graphs with nearly equitable 2-edge-colorings by proving the existence of a monochromatic cycle cover. Furthermore, we provide a polynomial-time algorithm that constructs a monochromatic cycle cover in complete graphs with an equitable 2-edge-coloring except possibly in a complete graph with four vertices. Our algorithm also finds a monochromatic cycle cover in complete graphs with a nearly equitable 2-edge-coloring except some special cases.

cs.DM↗

arXiv2021

Triangle-Free Equimatchable Graphs

A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least $5$. In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least $4$, i.e., equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al. in [1], which proves that the only connected triangle-free equimatchable $r$-regular graphs are $C_5$, $C_7$ and $K_{r,r}$, where $r$ is a positive integer. Given a non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.

cs.DM↗

arXiv2020

Total $k$-Uniform Graphs

A sequence of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v$ in the sequence has a neighbor which is adjacent to no vertex preceding $v$ in the sequence, and at the end every vertex of $G$ has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of $G$ (denoted by $γ_t(G)$) and the Grundy total domination number of $G$ (denoted by $γ_{gr}^t(G)$), respectively. In this paper, we study graphs with equal total domination number and Grundy total domination number. For every positive integer $k$, we call $G$ a total $k$-uniform graph if $γ_t(G)=γ_{gr}^t(G)=k$. We prove that there is no total $k$-uniform graph when $k$ is odd. In addition, we present a total 4-uniform graph which stands as a counterexample for a conjecture by T. Gologranc et.al. and provide a connected total 8-uniform graph. Moreover, we prove that every total $k$-uniform, connected and false twin-free graph is regular for every even $k$. We also show that there is no total $k$-uniform chordal connected graph with $k\geq 4$ and characterize all total $k$-uniform chordal graphs.

math.CO↗

arXiv2020

Well-Totally-Dominated Graphs

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.

math.CO↗

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↗

arXiv2019

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is the smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. % In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to that of finding an $r$-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of $r=2$, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of $r$ and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT algorithm.

cs.CC↗

arXiv2018

Mind the Independence Gap

The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most $k$ is polynomial-time solvable if $k$ is fixed and co-NP-complete if $k$ is part of input. We also investigate the complexity of the independent set problem in graph classes related to independence gap, showing that the problem is NP-complete in the class of graphs of independence gap at most one and polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomial time in the class of $\{$claw, 2$P_3\}$-free graphs.

math.CO↗

arXiv2018

On Almost Well-Covered Graphs of Girth at Least 6

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.

cs.DM↗

arXiv2018

Equimatchable Claw-Free Graphs

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide, to the best of our knowledge, the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.

cs.DM↗

arXiv2018

On a Class of Graphs with Large Total Domination Number

Let $γ(G)$ and $γ_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $γ_t(G) \leq 2γ(G)$. We provide a characterization of a large family of graphs (including chordal graphs) satisfying $γ_t(G)= 2γ(G)$, strictly generalizing the results of Henning (2001) and Hou et al. (2010), and partially answering an open question of Henning (2009).

math.CO↗