arXiv subjects

Tuomo Lehtilä

Publications and source records attributed to Tuomo Lehtilä.

25 records · Page 2Linked to original sources

arXiv2022

The RED-BLUE SEPARATION problem on graphs

We introduce the Red-Blue Separation problem on graphs, where we are given a graph $G=(V,E)$ whose vertices are colored either red or blue, and we want to select a (small) subset $S \subseteq V$, called red-blue separating set, such that for every red-blue pair of vertices, there is a vertex $s \in S$ whose closed neighborhood contains exactly one of the two vertices of the pair. We study the computational complexity of Red-Blue Separation, in which one asks whether a given red-blue colored graph has a red-blue separating set of size at most a given integer. We prove that the problem is NP-complete even for restricted graph classes. We also show that it is always approximable in polynomial time within a factor of $2\ln n$, where $n$ is the input graph's order. In contrast, for triangle-free graphs and for graphs of bounded maximum degree, we show that Red-Blue Separation is solvable in polynomial time when the size of the smaller color class is bounded by a constant. However, on general graphs, we show that the problem is $W[2]$-hard even when parameterized by the solution size plus the size of the smaller color class. We also consider the problem Max Red-Blue Separation where the coloring is not part of the input. Here, given an input graph $G$, we want to determine the smallest integer $k$ such that, for every possible red-blue coloring of $G$, there is a red-blue separating set of size at most $k$. We derive tight bounds on the cardinality of an optimal solution of Max Red-Blue Separation, showing that it can range from logarithmic in the graph order, up to the order minus one. We also give bounds with respect to related parameters. For trees however we prove an upper bound of two-thirds the order. We then show that Max Red-Blue Separation is NP-hard, even for graphs of bounded maximum degree, but can be approximated in polynomial time within a factor of $O(\ln^2 n)$.

cs.DM↗

arXiv2022

Revisiting and improving upper bounds for identifying codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\ell)/2$, where $n$ is the order and $\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $\frac{5n+2\ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars.

math.CO↗

arXiv2022

Super Domination: Graph Classes, Products and Enumeration

The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given simple graph $G=(V,E)$, a dominating set of $G$ is a subset $S\subseteq V$ such that every vertex in $ V \setminus S$ is adjacent to at least one vertex in $S$. Furthermore, the DSP is the problem of finding a minimum-size dominating set and the corresponding minimum size, the domination number of $G$. In this, work we investigate a variant of the DSP, the super dominating set problem (SDSP), which has attracted much attention during the last years. A dominating set $S$ is called a super dominating set of $G$, if for every vertex $u\in \overline{S}=V \setminus S$, there exists a $v\in S$ such that $N(v)\cap \overline{S}=\{u\}$. Analogously, the SDSP is to find a minimum-size super dominating set, and the corresponding minimum size, the super domination number of $G$. The decision variants of both the DSP and the SDSP have shown to be $\mathcal{NP}$-hard. In this paper, we present tight bounds for the super domination number of the neighbourhood corona product, $r$-gluing, and the Hajós sum of two graphs. Additionally, we present infinite families of graphs attaining our bounds. Finally, we give the exact number of minimum size super dominating sets for some graph classes. In particular, the number of super dominating sets for cycles has quite surprising properties as it varies between values of the set $\{4,n,2n,\frac{5n^2-10n}{8}\}$ based on $n\mod4$.

math.CO↗

arXiv2022

Locating-dominating sets: from graphs to oriented graphs

A locating-dominating set in an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, we have $N(u)\cap S\ne N(v)\cap S$. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set $S$ such that for every $w\in V$, $N[w]^-\cap S=\emptyset$ and for each pair of vertices $u,v\in V\setminus S$, $N^-(u)\cap S\ne N^-(v)\cap S$. We consider the following two parameters. Given an undirected graph $G$, we look for $\overset{\rightarrow}γ_{LD}(G)$ ($\overset{\rightarrow}Γ_{LD}(G))$ which is the size of the smallest (largest) optimal locating-dominating set over all orientations of $G$. In particular, if $D$ is an orientation of $G$, then $\overset{\rightarrow}γ_{LD}(G)\leqγ_{LD}(D)\leq\overset{\rightarrow}Γ_{LD}(G)$. For the best orientation, we prove that, for every twin-free graph $G$ on $n$ vertices, $\overset{\rightarrow}γ_{LD}(G)\le n/2$ proving a ``directed version'' of a conjecture on $γ_{LD}(G)$. Moreover, we give some bounds for $\overset{\rightarrow}γ_{LD}(G)$ on many graph classes and drastically improve the value $n/2$ for (almost) $d$-regular graphs by showing that $\overset{\rightarrow}γ_{LD}(G)\in O(\log d/d\cdot n)$ using a probabilistic argument. While $\overset{\rightarrow}γ_{LD}(G)\leqγ_{LD}(G)$ holds for every graph $G$, we give some graph classes graphs for which $\overset{\rightarrow}Γ_{LD}(G)\geqγ_{LD}(G)$ and some for which $\overset{\rightarrow}Γ_{LD}(G)\leq γ_{LD}(G)$. We also give general bounds for $\overset{\rightarrow}Γ_{LD}(G)$. Finally, we show that for many graph classes $\overset{\rightarrow}Γ_{LD}(G)$ is polynomial on $n$ but we leave open the question whether there exist graphs with $\overset{\rightarrow}Γ_{LD}(G)\in O(\log n)$.

math.CO↗

arXiv2021

Improved Lower Bound for Locating-Dominating Codes in Binary Hamming Spaces

In this article, we study locating-dominating codes in binary Hamming spaces $\mathbb{F}^n$. Locating-dominating codes have been widely studied since their introduction in 1980s by Slater and Rall. They are dominating sets suitable for distinguishing vertices in graphs. Dominating sets as well as locating-dominating codes have been studied in Hamming spaces in multiple articles. Previously, Honkala et al. (2004) have presented a lower bound for locating-dominating codes in binary Hamming spaces. In this article, we improve the lower bound for all values $n\geq10$. In particular, when $n=11$, we manage to improve the previous lower bound from $309$ to $317$. This value is very close to the current best known upper bound of $320$.

math.CO↗

arXiv2019

On Stronger Types of Locating-dominating Codes

Locating-dominating codes in a graph find their application in sensor networks and have been studied extensively over the years. A locating-dominating code can locate one object in a sensor network, but if there is more than one object, it may lead to false conclusions. In this paper, we consider stronger types of locating-dominating codes which can locate one object and detect if there are multiple objects. We study the properties of these codes and provide bounds on the smallest possible size of these codes, for example, with the aid of the Dilworth number and Sperner families. Moreover, these codes are studied in trees and Cartesian products of graphs. We also give the complete realization theorems for the coexistence of the smallest possible size of these codes and the optimal locating-dominating codes in a graph.

math.CO↗

arXiv2018

Solving a Conjecture on Identification in Hamming Graphs

Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In particular, there are a lot of results regarding the binary hypercubes, that is, the Hamming graphs $K_2^n$. In 2008, Gravier et al. started investigating identification in $K_q^2$. Goddard and Wash, in 2013, studied identifying codes in the general Hamming graphs $K_q^n$. They stated, for instance, that $γ^{ID}(K_q^n)\leq q^{n-1}$ for any $q$ and $n\geq3$. Moreover, they conjectured that $γ^{ID}(K_q^3)=q^2$. In this article, we show that $γ^{ID}(K_q^3)\leq q^2-q/4$ when $q$ is a power of four, disproving the conjecture. Our approach is based on the recursive use of suitable designs. Goddard and Wash also gave the following lower bound $γ^{ID}(K_q^3)\ge q^2-q\sqrt{q}$. We improve this bound to $γ^{ID}(K_q^3)\ge q^2-\frac{3}{2} q$. The conventional methods used for obtaining lower bounds on identifying codes do not help here. Hence, we provide a different technique building on the approach of Goddard and Wash. Moreover, we improve the above mentioned bound $γ^{ID}(K_q^n)\leq q^{n-1}$ to $γ^{ID}(K_q^n)\leq q^{n-k}$ for $n=3\frac{q^k-1}{q-1}$ when $q$ is a prime power. For this bound, we utilize suitable linear codes over finite fields and a class of closely related codes, namely, the self-locating-dominating codes. In addition, we show that the self-locating-dominating codes satisfy the result $γ^{SLD}(K_q^3)=q^2$ related to the above conjecture.

math.CO↗