arXiv subjects

Magdalena Lemańska

Publications and source records attributed to Magdalena Lemańska.

11 recordsLinked to original sources

arXiv2026

Isolation subdivision number of a graph

For a graph $G=(V,E),$ a set $S \subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by $ι(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\rm sd}_ι(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than $ι(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\leq {\rm sd}_ι(T)\leq 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\rm sd}_ι(T)=1.$

math.CO↗

arXiv2026

Isolation critical graphs under multiple edge subdivision

This paper introduces the notion of an $(ι,q)$-critical graph. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the subgraph induced by the set of vertices that are not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical if every subdivision of $q$ edges of $G$ gives a graph whose isolation number is greater than $ι(G)$, and $G$ has $q-1$ edges such that subdividing them gives a graph whose isolation number is $ι(G)$. We show that an $(ι,q)$-critical graph exists for every integer $q \ge 1$. We prove that if $G$ is a connected $m$-edge non-star graph, then $G$ is $(ι,q)$-critical for some $q \le m - 1$. We show that this bound is best possible. We provide a general characterization of $(ι,1)$-critical graphs as well as a constructive characterization of $(ι,1)$-critical trees, demonstrating that $(ι,1)$-criticality can be checked in linear time for trees.

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arXiv2025

Upper bounds on the $k$-isolation number

The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $ι(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the closed neighbourhood $N[D]$ of $D$ from $G$) has no edges. For $k \geq 1$, the $k$-isolation number of $G$ is the size of a smallest subset $D$ of $V(G)$ such that the maximum degree of $G-N[D]$ is at most $k-1$. Thus, $ι_1(G) = ι(G)$. Let $n$ and $\ell$ be the number of vertices and the number of leaves of $G$, respectively. We show that if $n \geq 3$ and $G$ is connected, then $ι_k(G) \leq \frac{n - \ell}{2}$. We also show that if $G$ is a tree $T$, then $ι(T) \leq \frac{n + \ell}{4}$ and $ι_k(T) \leq \frac{n + \ell}{2k+1}$ for $k \geq 2$. These bounds together improve the inequality $ι_k(T) \leq \frac{n}{k+2}$ of Caro and Hansberg except that their inequality is better if $k \geq 2$ and $\frac{k-1}{k+2}n < \ell < \frac{k}{k+2}n$. Each of the new bounds is attainable if it is an integer. For each of them, we characterize all the graphs that attain it.

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arXiv2019

On the connected and weakly convex domination numbers

In this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.

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arXiv2017

Graphs with equal domination and certified domination numbers

A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $γ(G)$ ($Γ(G)$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $G$. A subset $D\subseteq V_G$ is called a certified dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ has either zero or at least two neighbors in $V_G-D$. The cardinality of a~smallest (largest minimal, respectively) certified dominating set of $G$ is called the certified upper certified, respectively domination number of $G$ and is denoted by $γ_{\rm cer}(G)$ ($Γ_{\rm cer}(G)$, respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.

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arXiv2017

A characterization of trees having a minimum vertex cover which is also a minimum total dominating set

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(γ_t-τ)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(γ_t-τ)$-set.

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arXiv2016

Certified domination

Imagine that we are given a set $D$ of officials and a set $W$ of civils. For each civil $x \in W$, there must be an official $v \in D$ that can serve $x$, and whenever any such $v$ is serving $x$, there must also be another civil $w \in W$ that observes $v$, that is, $w$ may act as a kind of witness, to avoid any abuse from $v$. What is the minimum number of officials to guarantee such a service, assuming a given social network? In this paper, we introduce the concept of certified domination that perfectly models the aforementioned problem. Specifically, a dominating set $D$ of a graph $G=(V_G,E_G)$ is said to be certified if every vertex in $D$ has either zero or at least two neighbours in $V_G\setminus D$. The cardinality of a minimum certified dominating set in $G$ is called the certified domination number of $G$. Herein, we present the exact values of the certified domination number for some classes of graphs as well as provide some upper bounds on this parameter for arbitrary graphs. We then characterise a wide class of graphs with equal domination and certified domination numbers and characterise graphs with large values of certified domination numbers. Next, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. We also provide Nordhaus-Gaddum type inequalities for the certified domination number. Finally, we show that the (decision) certified domination problem is NP-complete.

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arXiv2016

Coronas and domination subdivision number of a graph

In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.

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arXiv2014

Relations between edge removing and edge subdivision concerning domination number of a graph

Let $e$ be an edge of a connected simple graph $G$. The graph obtained by removing (subdividing) an edge $e$ from $G$ is denoted by $G-e$ ($G_e$). As usual, $γ(G)$ denotes the domination number of $G$. We call $G$ an SR-graph if $γ(G-e) = γ(G_e)$ for any edge $e$ of $G$, and $G$ is an ASR-graph if $γ(G - e) \neq (G_e)$ for any edge $e$ of $G$. In this work we give several examples of SR and ASR-graphs. Also, we characterize SR-trees and show that ASR-graphs are $γ$-insensitive.

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arXiv2013

Total domination multisubdivision number of a graph

The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define the total domination multisubdivision number msd$_{γ_t}(G)$ of a graph $G$ and we show that for any connected graph $G$ of order at least two, msd$_{γ_t}(G)\leq 3.$ We show that for trees the total domination multisubdivision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees $T$ with msd$_{γ_t}(T)=1$.

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arXiv2013

Graphs with $C_3_-free vertices are not universal fixers

A non-isolated vertex $x\in V(G)$ is called $C_{3}$-free if $x$ belongs to no triangle of $G$. In \cite{BMW} Burger, Mynhardt and Weakley introduced the idea of universal fixers. Let $G=(V,E)$ be a graph with $n$ vertices and $G'$ a copy of $G$. For a bijective function $π:V(G)\mapsto V (G')$, we define the prism $πG$ of $G$ as follows: $V(πG)=V(G)\cup V(G')$ and $E(πG)=E(G)\cup E(G')\cup M_π$, where $M_π=\{uπ(u): u\in V(G)\}$. Let $γ(G)$ be the domination number of $G$. If $γ(πG)=γ(G)$ for any bijective function $π$, then $G$ is called a universal fixer. In \cite{MX} it is conjectured that the only universal fixer is the edgeless graph $\bar{K_n}$. In this note, we prove that any graph $G$ with $C_3$-free vertices is not a universal fixer graph.

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