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Dieter Rautenbach

Publications and source records attributed to Dieter Rautenbach.

At least 55 records · Page 3Linked to original sources
arXiv2021

Zero-sum copies of spanning forests in zero-sum complete graphs

For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{ -1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$ of $F$ in $K_n$. In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$. We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq Δ(F)+1$, where $Δ(F)$ is the maximum degree of $F$. We conjecture that this bound can be improved to $|c(E(F'))|\leq (Δ(F)-1)/2$ and verify this for $F$ being the star $K_{1,n-1}$. Under some simple necessary divisibility conditions, we show the existence of a zero-sum $P_3$-factor, and, for sufficiently large $n$, also of a zero-sum $P_4$-factor.

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arXiv2020

Minimum distance-unbalancedness of trees

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. Confirming one of their conjectures, we show that the stars minimize the distance-unbalancedness among all trees of a fixed order.

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arXiv2020

Low Weight Perfect Matchings

Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer $n$ and every function $σ\colon E(K_{4n})\to\{-1,1\}$ with $σ\left(E(K_{4n})\right)=0$, there is a perfect matching $M$ in $K_{4n}$ with $σ(M)=0$. Strengthening a result of Caro and Yuster, we show that for every positive integer $n$ and every function $σ\colon E(K_{4n})\to\{-1,1\}$ with $\left|σ\left(E(K_{4n})\right)\right|<n^2+11n+2,$ there is a perfect matching $M$ in $K_{4n}$ with $|σ(M)|\leq 2$. Both these results are best possible.

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arXiv2020

Exponential Independence in Subcubic Graphs

A set $S$ of vertices of a graph $G$ is exponentially independent if, for every vertex $u$ in $S$, $$\sum\limits_{v\in S\setminus \{ u\}}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}<1,$$ where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u$ and $v$ in the graph $G-(S\setminus \{ u,v\})$. The exponential independence number $α_e(G)$ of $G$ is the maximum order of an exponentially independent set in $G$. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order $n$ have exponentially independent sets of order $Ω(n/\log^2(n))$, that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order $n$ have exponentially independent sets of order $(n+3)/4$.

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arXiv2020

Reconfiguring dominating sets in minor-closed graph classes

For a graph $G$, two dominating sets $D$ and $D'$ in $G$, and a non-negative integer $k$, the set $D$ is said to $k$-transform to $D'$ if there is a sequence $D_0,\ldots,D_\ell$ of dominating sets in $G$ such that $D=D_0$, $D'=D_\ell$, $|D_i|\leq k$ for every $i\in \{ 0,1,\ldots,\ell\}$, and $D_i$ arises from $D_{i-1}$ by adding or removing one vertex for every $i\in \{ 1,\ldots,\ell\}$. We prove that there is some positive constant $c$ and there are toroidal graphs $G$ of arbitrarily large order $n$, and two minimum dominating sets $D$ and $D'$ in $G$ such that $D$ $k$-transforms to $D'$ only if $k\geq \max\{ |D|,|D'|\}+c\sqrt{n}$. Conversely, for every hereditary class ${\cal G}$ that has balanced separators of order $n\mapsto n^α$ for some $α<1$, we prove that there is some positive constant $C$ such that, if $G$ is a graph in ${\cal G}$ of order $n$, and $D$ and $D'$ are two dominating sets in $G$, then $D$ $k$-transforms to $D'$ for $k=\max\{ |D|,|D'|\}+\lfloor Cn^α\rfloor$.

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arXiv2020

Biholes in balanced bipartite graphs

A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $|I\cap A|=|I\cap B|$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving the vertex degrees and the average degree.

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arXiv2020

Additive Tree $O(ρ\log n)$-Spanners from Tree Breadth $ρ$

The tree breadth ${\rm tb}(G)$ of a connected graph $G$ is the smallest non-negative integer $ρ$ such that $G$ has a tree decomposition whose bags all have radius at most $ρ$. We show that, given a connected graph $G$ of order $n$ and size $m$, one can construct in time $O(m\log n)$ an additive tree $O\big({\rm tb}(G)\log n\big)$-spanner of $G$, that is, a spanning subtree $T$ of $G$ in which $d_T(u,v)\leq d_G(u,v)+O\big({\rm tb}(G)\log n\big)$ for every two vertices $u$ and $v$ of $G$. This improves earlier results of Dragan and Köhler (Algorithmica 69 (2014) 884-905), who obtained a multiplicative error of the same order, and of Dragan and Abu-Ata (Theoretical Computer Science 547 (2014) 1-17), who achieved the same additive error with a collection of $O(\log n)$ trees.

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arXiv2020

Acyclic matchings in graphs of bounded maximum degree

A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $Δ$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{Δ^2},$ and we explain how to find such an acyclic matching in polynomial time.

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arXiv2019

Cubic graphs with equal independence number and matching number

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $δ(G) α(G)\leq Δ(G) μ(G)$ for every graph $G$ with minimum degree $δ(G)$, maximum degree $Δ(G)$, independence number $α(G)$, and matching number $μ(G)$. Answering some problems they posed, we characterize the extremal graphs for $δ(G)<Δ(G)$ as well as for $δ(G)=Δ(G)=3$.

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arXiv2019

Domination versus edge domination

We propose the conjecture that the domination number $γ(G)$ of a $Δ$-regular graph $G$ with $Δ\geq 1$ is always at most its edge domination number $γ_e(G)$, which coincides with the domination number of its line graph. We prove that $γ(G)\leq \left(1+\frac{2(Δ-1)}{Δ2^Δ}\right)γ_e(G)$ for general $Δ\geq 1$, and $γ(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)γ_e(G)$ for $Δ=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.

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arXiv2019

On the Computational Complexity of the Bipartizing Matching Problem

We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits a~$(2,1)$-coloring, which is a~$2$-coloring of~$V(G)$ such that each color class induces a graph of maximum degree at most~$1$. We determine a dichotomy related to the~{\sf NP}-completeness of this problem, where we show that it is~{\sf NP}-complete even for $3$-colorable planar graphs of maximum degree~$4$, while it is known that the problem can be solved in polynomial time for graphs of maximum degree at most~$3$. In addition we present polynomial-time algorithms for some graph classes, including graphs in which every odd cycle is a triangle, graphs of small dominating sets, and~$P_5$-free graphs. Additionally, we show that the problem is fixed parameter tractable when parameterized by the clique-width, which implies polynomial-time solution for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs. Finally, an~$O\left(2^{O\left(vc(G)\right)} \cdot n\right)$-time algorithm and a kernel of at most~$2\cdot nd(G)$ vertices are presented, where~$vc(G)$ and~$nd(G)$ are the vertex cover number and the neighborhood diversity of~$G$, respectively.

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arXiv2019

Bounding and approximating minimum maximal matchings in regular graphs

The edge domination number $γ_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on $γ_e(G)$ for regular and non-regular graphs $G$ in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

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arXiv2019

On the maximum number of minimum total dominating sets in forests

We propose the conjecture that every tree with order $n$ at least $2$ and total domination number $γ_t$ has at most $\left(\frac{n-\frac{γ_t}{2}}{\frac{γ_t}{2}}\right)^{\frac{γ_t}{2}}$ minimum total dominating sets. As a relaxation of this conjecture, we show that every forest $F$ with order $n$, no isolated vertex, and total domination number $γ_t$ has at most $\min\left\{\left(8\sqrt{e}\, \right)^{γ_t}\left(\frac{n-\frac{γ_t}{2}}{\frac{γ_t}{2}}\right)^{\frac{γ_t}{2}}, (1+\sqrt{2})^{n-γ_t},1.4865^n\right\}$ minimum total dominating sets.

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arXiv2018

Linear programming based approximation for unweighted induced matchings --- breaking the $Δ$ barrier

A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (Approximating weighted induced matchings, Discrete Applied Mathematics 243 (2018) 304-310) provide an approximation algorithm with ratio $Δ$ for the weighted version of the induced matching problem on graphs of maximum degree $Δ$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $Δ-1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}Δ+O(1)$, and that also the approximation ratio can be improved at least to this value. We provide primal-dual approximation algorithms with ratios $(1-ε) Δ+ \frac{1}{2}$ for general $Δ$ with $ε\approx 0.02005$, and $\frac{7}{3}$ for $Δ=3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

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arXiv2018

Uniquely restricted matchings in subcubic graphs without short cycles

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1) / 3$ except for two exceptional cubic graphs of order $14$ and $20$.

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arXiv2018

Relating broadcast independence and independence

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $α_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. Clearly, $α_b(G)$ is at least the independence number $α(G)$ for every connected graph $G$. Our main result implies $α_b(G)\leq 4α(G)$. We prove a tight inequality and characterize all extremal graphs.

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arXiv2018

Girth, minimum degree, independence, and broadcast independence

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $α_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. It is known that $α(G)\leq α_b(G)\leq 4α(G)$ for every connected graph $G$, where $α(G)$ is the independence number of $G$. If $G$ has girth $g$ and minimum degree $δ$, we show that $α_b(G)\leq 2α(G)$ provided that $g\geq 6$ and $δ\geq 3$ or that $g\geq 4$ and $δ\geq 5$. Furthermore, we show that, for every positive integer $k$, there is a connected graph $G$ of girth at least $k$ and minimum degree at least $k$ such that $α_b(G)\geq 2\left(1-\frac{1}{k}\right)α(G)$. Our results imply that lower bounds on the girth and the minimum degree of a connected graph $G$ can lower the fraction $\frac{α_b(G)}{α(G)}$ from $4$ below $2$, but not any further.

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arXiv2018

Algorithmic aspects of broadcast independence

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $α_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. We describe an efficient algorithm that determines the broadcast independence number of a given tree. Furthermore, we show NP-hardness of the broadcast independence number for planar graphs of maximum degree four, and hardness of approximation for general graphs. Our results solve problems posed by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi (2006), Hedetniemi (2006), and Ahmane, Bouchemakh, Sopena (2018).

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