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Dieter Rautenbach

Publications and source records attributed to Dieter Rautenbach.

At least 91 records · Page 5Linked to original sources
arXiv2017

How to determine if a random graph with a fixed degree sequence has a giant component

For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $λ(n)$ for some function $λ$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$ has a giant component and the probability that $G(\mathcal{D})$ has no giant component are bounded away from $1$.

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arXiv2016

Uniquely restricted matchings and edge colorings

A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs. In particular, we achieve a ratio of $9/5$ for subcubic bipartite graphs, improving over a $2$-approximation algorithm proposed by Mishra. Furthermore, we study the uniquely restricted chromatic index of a graph, defined as the minimum number of uniquely restricted matchings into which its edge set can be partitioned. We provide tight upper bounds in terms of the maximum degree and characterize all extremal graphs. Our constructive proofs yield efficient algorithms to determine the corresponding edge colorings.

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arXiv2016

Large Values of the Clustering Coefficient

A prominent parameter in the context of network analysis, originally proposed by Watts and Strogatz (Collective dynamics of `small-world' networks, Nature 393 (1998) 440-442), is the clustering coefficient of a graph $G$. It is defined as the arithmetic mean of the clustering coefficients of its vertices, where the clustering coefficient of a vertex $u$ of $G$ is the relative density $m(G[N_G(u)])/{d_G(u)\choose 2}$ of its neighborhood if $d_G(u)$ is at least $2$, and $0$ otherwise. It is unknown which graphs maximize the clustering coefficient among all connected graphs of given order and size. We determine the maximum clustering coefficients among all connected regular graphs of a given order, as well as among all connected subcubic graphs of a given order. In both cases, we characterize all extremal graphs. Furthermore, we determine the maximum increase of the clustering coefficient caused by adding a single edge.

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arXiv2016

Bounds on the Burning Number

Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato, Janssen, and Roshanbin [Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22] define the burning number $b(G)$ of a graph $G$ as the smallest integer $k$ for which there are vertices $x_1,\ldots,x_k$ such that for every vertex $u$ of $G$, there is some $i\in \{ 1,\ldots,k\}$ with ${\rm dist}_G(u,x_i)\leq k-i$, and ${\rm dist}_G(x_i,x_j)\geq j-i$ for every $i,j\in \{ 1,\ldots,k\}$. For a connected graph $G$ of order $n$, they prove that $b(G)\leq 2\left\lceil\sqrt{n}\right\rceil-1$, and conjecture $b(G)\leq \left\lceil\sqrt{n}\right\rceil$. We show that $b(G)\leq \sqrt{\frac{32}{19}\cdot \frac{n}{1-ε}}+\sqrt{\frac{27}{19ε}}$ and $b(G)\leq \sqrt{\frac{12n}{7}}+3\approx 1.309 \sqrt{n}+3$ for every connected graph $G$ of order $n$ and every $0<ε<1$. For a tree $T$ of order $n$ with $n_2$ vertices of degree $2$, and $n_{\geq 3}$ vertices of degree at least $3$, we show $b(T)\leq \left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil$ and $b(T)\leq \left\lceil\sqrt{n}\right\rceil+n_{\geq 3}$. Furthermore, we characterize the binary trees of depth $r$ that have burning number $r+1$.

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arXiv2016

Extremal Values of the Chromatic Number for a Given Degree Sequence

For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $χ_{\min}(d)$ and the largest chromatic number $χ_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then $χ_{\min}(d)\leq \max\left\{ 3,d_1-\frac{n+1}{4d_1}+4\right\}$, and, if $\sqrt{n+\frac{1}{4}}-\frac{1}{2}>d_1\geq d_n\geq 1$, then $χ_{\max}(d)=\max\limits_{i\in [n]}\min\left\{ i,d_i+1\right\}$. For a given degree sequence $d$ with bounded entries, we show that $χ_{\min}(d)$, $χ_{\max}(d)$, and also the smallest independence number $α_{\min}(d)$ among all graphs with degree sequence $d$, can be determined in polynomial time.

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arXiv2016

Some Comments on the Slater number

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $γ(G)$ of $G$. We show that deciding the equality of $γ(G)$ and $s\ell(G)$ for a given graph $G$ is NP-complete but that one can decide efficiently whether $γ(G)>s\ell(G)$ or $γ(G)\leq \left(\left\lceil\ln \left(\frac{n(G)}{s\ell(G)}\right)\right\rceil+1\right)s\ell(G)$. For real numbers $α$ and $β$ with $α\geq \max\{ 0,β\}$, let ${\cal G}(α,β)$ be the class of non-null graphs $G$ such that every non-null subgraph $H$ of $G$ has at most $αn(H)-β$ many edges. Generalizing a result of Desormeaux, Haynes, and Henning, we show that $γ(G)\leq (2α+1)s\ell(G)-2β$ for every graph $G$ in ${\cal G}(α,β)$ with $α\leq \frac{3}{2}$. Furthermore, we show that $γ(G)/s\ell(G)$ is bounded for graphs $G$ in ${\cal G}(α,β)$ if and only if $α<2$. For an outerplanar graph $G$ with $s\ell(G)\geq 2$, we show $γ(G)\leq 6s\ell(G)-6$. In analogy to $s\ell(G)$, we propose $s\ell_t(G)=\min\{ s: d_1+\cdots+d_s\geq n\}$ as a lower bound on the total domination number. Strengthening results due to Raczek as well as Chellali and Haynes, we show that $s\ell_t(T)\geq \frac{n+2-n_1}{2}$ for every tree $T$ of order $n$ at least $2$ with $n_1$ endvertices.

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arXiv2016

Some Bounds on the Zero Forcing Number of a Graph

A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if initially labeling all vertices in $Z$ with $1$ and all remaining vertices of $G$ with $0$, and then, iteratively and as long as possible, changing the label of some vertex $u$ from $0$ to $1$ if $u$ is the only neighbor with label $0$ of some vertex with label $1$, results in the entire vertex set of $G$. The zero forcing number $Z(G)$, defined as the minimum order of a zero forcing set of $G$, was proposed as an upper bound of the corank of matrices associated with $G$, and was also considered in connection with quantum physics and logic circuits. In view of the computational hardness of the zero forcing number, upper and lower bounds are of interest. Refining results of Amos, Caro, Davila, and Pepper, we show that $Z(G)\leq \frac{Δ-2}{Δ-1}n$ for a connected graph $G$ of order $n$ and maximum degree $Δ$ at least $3$ if and only if $G$ does not belong to $\{ K_{Δ+1},K_{Δ,Δ},K_{Δ-1,Δ},G_1,G_2\}$, where $G_1$ and $G_2$ are two specific graphs of orders $5$ and $7$, respectively. For a connected graph $G$ of order $n$, maximum degree $3$, and girth at least $5$, we show $Z(G)\leq \frac{n}{2}-Ω\left(\frac{n}{\log n}\right)$. Using a probabilistic argument, we show $Z(G)\leq \left(1-\frac{H_r}{r}+o\left(\frac{H_r}{r}\right)\right)n$ for an $r$-regular graph $G$ of order $n$ and girth at least $5$, where $H_r$ is the $r$-th harmonic number. Finally, we show $Z(G)\geq (g-2)(δ-2)+2$ for a graph $G$ of girth $g\in \{ 5,6\}$ and minimum degree $δ$, which partially confirms a conjecture of Davila and Kenter.

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arXiv2016

Exponential Independence

For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V(G)\setminus S$, and a vertex $v$ in $S$, let ${\rm dist}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G-(S\setminus \{ v\})$. Dankelmann et al. (Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}(u)\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where $w_{(G,S)}(u)=\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}$. Inspired by this notion, we define $S$ to be an exponential independent set of $G$ if $w_{(G,S\setminus \{ u\})}(u)<1$ for every vertex $u$ in $S$, and the exponential independence number $α_e(G)$ of $G$ as the maximum order of an exponential independent set of $G$. Similarly as for exponential domination, the non-local nature of exponential independence leads to many interesting effects and challenges. Our results comprise exact values for special graphs as well as tight bounds and the corresponding extremal graphs. Furthermore, we characterize all graphs $G$ for which $α_e(H)$ equals the independence number $α(H)$ for every induced subgraph $H$ of $G$, and we give an explicit characterization of all trees $T$ with $α_e(T)=α(T)$.

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arXiv2016

Relating Domination, Exponential Domination, and Porous Exponential Domination

The domination number $γ(G)$ of a graph $G$, its exponential domination number $γ_e(G)$, and its porous exponential domination number $γ_e^*(G)$ satisfy $γ_e^*(G)\leq γ_e(G)\leq γ(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $γ_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $γ_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show $γ_{e,f}^*(T)=\frac{n+2}{6}$ and $γ_e(T)\leq 2γ_{e,f}^*(T)$. We characterize the two classes of subcubic trees $T$ with $γ_e(T)=γ_{e,f}^*(T)$ and $γ(T)=γ_e(T)$, respectively. Using linear programming arguments, we establish several lower bounds on the fractional porous exponential domination number in more general settings.

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arXiv2016

Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees

Let $G$ be a graph, and let $ρ\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_ρ(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\lceil ρd_G(u)\rceil$ neighbors in it. If $H_ρ(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\mapsto \lceil ρd_G(u)\rceil$. Let $h_ρ(G)$ be the minimum cardinality of such an irreversible dynamic monopoly. For a connected graph $G$ of maximum degree at least $\frac{1}ρ$, Chang (Triggering cascades on undirected connected graphs, Information Processing Letters 111 (2011) 973-978) showed $h_ρ(G)\leq 5.83ρn(G)$, which was improved by Chang and Lyuu (Triggering cascades on strongly connected directed graphs, Theoretical Computer Science 593 (2015) 62-69) to $h_ρ(G)\leq 4.92ρn(G)$. We show that for every $ε>0$, there is some $ρ(ε)>0$ such that $h_ρ(G) \leq(2+ε)ρn(G)$ for every $ρ$ in $(0,ρ(ε))$, and every connected graph $G$ that has maximum degree at least $\frac{1}ρ$ and girth at least $5$. Furthermore, we show that $h_ρ(T) \leq ρn(T)$ for every $ρ$ in $(0,1]$, and every tree $T$ that has order at least $\frac{1}ρ$.

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arXiv2015

Relating $2$-Rainbow Domination to Roman domination

For a graph $G$, let $γ_R(G)$ and $γ_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $γ_{r2}(G)\leq γ_R(G)\leq \frac{3}{2}γ_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $γ_R(G)-γ_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs efficiently. We show that for every fixed non-negative integer $k$, the recognition of the connected $K_4$-free graphs $G$ with $γ_R(G)-γ_{r2}(G)=k$ is NP-hard, which implies that there is most likely no good characterization of these graphs. We characterize the graphs $G$ such that $γ_{r2}(H)=γ_R(H)$ for every induced subgraph $H$ of $G$, and collect several properties of the graphs $G$ with $γ_R(G)=\frac{3}{2}γ_{r2}(G)$.

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arXiv2015

Dominating Sets inducing Large Components in Maximal Outerplanar Graphs

For a maximal outerplanar graph $G$ of order $n$ at least $3$, Matheson and Tarjan showed that $G$ has domination number at most $n/3$. Similarly, for a maximal outerplanar graph $G$ of order $n$ at least $5$, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that $G$ has total domination number at most $2n/5$ unless $G$ is isomorphic to one of two exceptional graphs of order $12$. We present a unified proof of a common generalization of these two results. For every positive integer $k$, we specify a set ${\cal H}_k$ of graphs of order at least $4k+4$ and at most $4k^2-2k$ such that every maximal outerplanar graph $G$ of order $n$ at least $2k+1$ that does not belong to ${\cal H}_k$ has a dominating set $D$ of order at most $\lfloor\frac{kn}{2k+1}\rfloor$ such that every component of the subgraph $G[D]$ of $G$ induced by $D$ has order at least $k$.

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arXiv2015

Exponential Domination in Subcubic Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $γ_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$. In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq γ_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $ε>0$, there is some $g$ such that $γ_e(G)\leq εn(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<α<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $γ_e(G)\leq \frac{3n(G)}{\ln(n(G))^α}$. If $T$ is a subcubic tree, then $γ_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $γ_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.

math.CO↗
arXiv2015

Bounds on the Exponential Domination Number

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $γ_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$. Dankelmann et al. show $$\frac{1}{4}({\rm d}+2)\leq γ_e(G)\leq \frac{2}{5}(n+2)$$ for a connected graph $G$ of order $n$ and diameter ${\rm d}$. We provide further bounds and in particular strengthen their upper bound. Specifically, for a connected graph $G$ of order $n$, maximum degree $Δ$ at least $3$, radius ${\rm r}$ at least $1$, we show \begin{eqnarray*} γ_e(G) & \geq & \left(\frac{n}{13(Δ-1)^2}\right)^{\frac{\log_2(Δ-1)+1}{\log_2^2(Δ-1)+\log_2(Δ-1)+1}},\\[3mm] γ_e(G) & \leq & 2^{2{\rm r}-2}\mbox{, and }\\[3mm] γ_e(G) & \leq & \frac{43}{108}(n+2). \end{eqnarray*}

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arXiv2015

Local Connectivity, Local Degree Conditions, some Forbidden Induced Subgraphs, and Cycle Extendability

The research in the present paper was motivated by the conjecture of Ryjáček that every locally connected graph is weakly pancyclic. For a connected locally connected graph $G$ of order at least $3$, our results are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is weakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycle extendable if and only if $2δ(G)\geq n(G)$. If $G$ is $\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$-free or $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$-free, then $G$ is fully cycle extendable. If $G$ is distinct from $K_1+K_1+\bar{K}_3$ and $\{ K_1+P_4,K_{1,4},K_2+(K_1\cup K_2)\}$-free, then $G$ is fully cycle extendable. Furthermore, if $G$ is a connected graph of order at least $3$ such that $$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$ for every induced path $vuw$ of order $3$ in $G$, then $G$ is fully cycle extendable, which implies that every connected locally Ore or locally Dirac graph of order at least $3$ is fully cycle extendable.

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arXiv2015

Averaging $2$-Rainbow Domination and Roman Domination

For a graph $G$, let $γ_{r2}(G)$ and $γ_R(G)$ denote the $2$-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) proved $γ_{r2}(G)+γ_R(G)\leq \frac{6}{4}n(G)$ for a connected graph $G$ of order $n(G)$ at least $3$. Furthermore, they conjectured $γ_{r2}(G)+γ_R(G)\leq \frac{4}{3}n(G)$ for a connected graph $G$ of minimum degree at least $2$ that is distinct from $C_5$. We characterize all extremal graphs for their inequality and prove their conjecture.

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arXiv2015

Relating $2$-rainbow domination to weak Roman domination

Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27-32) we prove $γ_{r2}(G)\leq 2γ_r(G)$ for every graph $G$, where $γ_{r2}(G)$ and $γ_r(G)$ denote the $2$-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{ K_4,K_4-e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.

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