A Note on Outer-Independent 2-Rainbow Domination in Graphs

Let G be a graph with vertex set V(G) and f:V(G)→{∅,{1},{2},{1,2}} be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: (i)V∅={x∈V(G):f(x)=∅} is an independent set of G. (ii)∪u∈N(v)f(u)={1,2} for every vertex v∈V∅. The outer-independent 2-rainbow domination number of G, denoted by γoir2(G), is the minimum weight ω(f)=∑x∈V(G)|f(x)| among all outer-independent 2-rainbow dominating functions f on G. In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β(G)≤γoir2(G)≤2β(G), where β(G) denotes the vertex cover number of G. Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain

무지개 집합 문제에서의 위상수학적 조합론

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅.$\mathcal{F}=\{S_1,\ldots,S_m\}$를 $V$의 공집합이 아닌 부분 집합들의 모임이라 할 때, $\mathcal{F}$의 무지개 집합이란 공집합이 아니며 $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$와 같은 형태로 주어지는 것으로 다음 조건을 만족하는 것을 말한다. $1\leq i_1<\cdots<i_k \leq m$이고 $j \ne j$이면 $s_{i_j} \ne s_{i_j'}$를 만족하며 각 $j \in [m]$에 대해 $s_{i_j} \in S_{i_j}$이다. 특히 $k=m$인 경우, 즉 모든 $S_i$들이 표현되면, 무지개 집합 $S$를 $\mathcal{F}$의 완전 무지개 집합이라고 한다. 주어진 집합계가 특정 조건을 만족하는 무지개 집합을 가지기 위한 충분 조건을 찾는 문제는 홀의 결혼 정리에서 시작되어 최근까지도 조합수학에서 가장 대표적 문제 중 하나로 여겨져왔다. 이러한 방향으로의 문제를 무지개 집합 문제라고 부른다. 본 학위논문에서는 무지개 집합 문제와 관련하여 위상수학적 홀의 정리와 위상수학적 다색 헬리 정리를 소개하고, (하이퍼)그래프에서의 무지개 덮개와 무지개 독립 집합에 관한 결과들을 다루고자 한다.Let $\mathcal{F}=\{S_1,\ldots,S_m\}$ be a finite family of non-empty subsets on the ground set $V$. A rainbow set of $\mathcal{F}$ is a non-empty set of the form $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$ with $1 \leq i_1 < \cdots < i_k \leq m$ such that $s_{i_j} \neq s_{i_{j'}}$ for every $j \neq j'$ and $s_{i_j} \in S_{i_j}$ for each $j \in [k]$. If $k = m$, namely if all $S_i$ is represented, then the rainbow set $S$ is called a full rainbow set of $\mathcal{F}$. Originated from the celebrated Hall's marriage theorem, it has been one of the most fundamental questions in combinatorics and discrete mathematics to find sufficient conditions on set-systems to guarantee the existence of certain rainbow sets. We call problems in this direction the rainbow set problems. In this dissertation, we give an overview on two topological tools on rainbow set problems, Aharoni and Haxell's topological Hall theorem and Kalai and Meshulam's topological colorful Helly theorem, and present some results on and rainbow independent sets and rainbow covers in (hyper)graphs.Abstract i 1 Introduction 1 1.1 Topological Hall theorem . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Topological colorful Helly theorem . . . . . . . . . . . . . . . . . 3 1.2.1 Collapsibility and Lerayness of simplicial complexes . . . 4 1.2.2 Nerve theorem and topological Helly theorem . . . . . . . 5 1.2.3 Topological colorful Helly theorem . . . . . . . . . . . . 6 1.3 Domination numbers and non-cover complexes of hypergraphs . . 7 1.3.1 Domination numbers of hypergraphs . . . . . . . . . . . . 10 1.3.2 Non-cover complexes of hypergraphs . . . . . . . . . . . . 10 1.4 Rainbow independent sets in graphs . . . . . . . . . . . . . . . . 12 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Collapsibility of non-cover complexes of graphs 16 2.1 The minimal exclusion sequences . . . . . . . . . . . . . . . . . . 16 2.2 Independent domination numbers and collapsibility numbers of non-cover complexes of graphs . . . . . . . . . . . . . . . . . . . 21 3 Domination numbers and non-cover complexes of hypergraphs 24 3.1 Proof of Theorem 1.3.4 . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Edge-annihilation . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Non-cover complexes for hypergraphs . . . . . . . . . . . 27 3.2 Lerayness of non-cover complexes . . . . . . . . . . . . . . . . . 30 3.2.1 Total domination numbers . . . . . . . . . . . . . . . . . 30 3.2.2 Independent domination numbers . . . . . . . . . . . . . 33 3.2.3 Edgewise-domination numbers . . . . . . . . . . . . . . . 34 3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Independent domination numbers of hypergraphs . . . . . 35 3.3.2 Independence complexes of hypergraphs . . . . . . . . . . 36 3.3.3 General position complexes . . . . . . . . . . . . . . . . . 37 3.3.4 Rainbow covers of hypergraphs . . . . . . . . . . . . . . 39 3.3.5 Collapsibility of non-cover complexes of hypergraphs . . . 40 4 Rainbow independent sets 42 4.1 Graphs avoiding certain induced subgraphs . . . . . . . . . . . . 42 4.1.1 Claw-free graphs . . . . . . . . . . . . . . . . . . . . . . 42 4.1.2 $\{C_4,C_5, . . . ,C_s\}$-free graphs . . . . . . . . . . . . . . . . . 44 4.1.3 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.4 $K_r$-free graphs and $K^{−}_r$-free graphs . . . . . . . . . . . . . 50 4.2 $k$-colourable graphs . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Graphs with bounded degrees . . . . . . . . . . . . . . . . . . . . 55 4.3.1 The case $m < n$ . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 A topological approach . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 67 Abstract (in Korean) 69 Acknowledgement (in Korean) 70Docto

Maximal 2-rainbow domination number of a graph

AbstractA 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v∈V(G) with f(v)=0̸ the condition ⋃u∈N(v)f(u)={1,2} is fulfilled, where N(v) is the open neighborhood of v. A maximal 2-rainbow dominating function on a graph G is a 2-rainbow dominating function f such that the set {w∈V(G)|f(w)=0̸} is not a dominating set of G. The weight of a maximal 2RDF f is the value ω(f)=∑v∈V|f(v)|. The maximal 2-rainbow domination number of a graph G, denoted by γmr(G), is the minimum weight of a maximal 2RDF of G. In this paper we initiate the study of maximal 2-rainbow domination number in graphs. We first show that the decision problem is NP-complete even when restricted to bipartite or chordal graphs, and then, we present some sharp bounds for γmr(G). In addition, we determine the maximal rainbow domination number of some graphs

Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

Further results on outer independent 22-rainbow dominating functions of graphs

Let $G=(V(G),E(G))$ be a graph. A function $f:V(G)\rightarrow \mathbb{P}(\{1,2\})$ is a $2$-rainbow dominating function if for every vertex $v$ with $f(v)=\emptyset$, f\big{(}N(v)\big{)}=\{1,2\}. An outer-independent $2$-rainbow dominating function (OI$2$RD function) of $G$ is a $2$-rainbow dominating function $f$ for which the set of all $v\in V(G)$ with $f(v)=\emptyset$ is independent. The outer independent $2$-rainbow domination number (OI$2$RD number) $\gamma_{oir2}(G)$ is the minimum weight of an OI$2$RD function of $G$. In this paper, we first prove that $n/2$ is a lower bound on the OI$2$RD number of a connected claw-free graph of order $n$ and characterize all such graphs for which the equality holds, solving an open problem given in an earlier paper. In addition, a study of this parameter for some graph products is carried out. In particular, we give a closed (resp. an exact) formula for the OI$2$RD number of rooted (resp. corona) product graphs and prove upper bounds on this parameter for the Cartesian product and direct product of two graphs

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

Domination parameters with number 2: interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure

Domination parameters with number 2: Interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin