arXiv subjects
Publications and source records attributed to Dieter Rautenbach.
We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.
We show that an identifying code of minimum order in the complementary prism of a cycle of order $n$ has order $7n/9+Θ(1)$. Furthermore, we observe that the clique-width of the complementary prism of a graph of clique-width $k$ is at most $4k$, and discuss some algorithmic consequences.
For a sequence $d$ of non-negative integers, let ${\cal G}(d)$ and ${\cal F}(d)$ be the sets of all graphs and forests with degree sequence $d$, respectively. Let $γ_{\min}(d)=\min\{ γ(G):G\in {\cal G}(d)\}$, $α_{\max}(d)=\max\{ α(G):G\in {\cal G}(d)\}$, $γ_{\min}^{\cal F}(d)=\min\{ γ(F):F\in {\cal F}(d)\}$, and $α_{\max}^{\cal F}(d)=\max\{ α(F):F\in {\cal F}(d)\}$ where $γ(G)$ is the domination number and $α(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $α_{\max}(d)$ can be determined in polynomial time. We establish the existence of realizations $G\in {\cal G}(d)$ with $γ_{\min}(d)=γ(G)$, and $F_γ,F_α\in {\cal F}(d)$ with $γ_{\min}^{\cal F}(d)=γ(F_γ)$ and $α_{\max}^{\cal F}(d)=α(F_α)$ that have strong structural properties. This leads to an efficient algorithm to determine $γ_{\min}(d)$ for every given degree sequence $d$ with bounded entries as well as closed formulas for $γ_{\min}^{\cal F}(d)$ and $α_{\max}^{\cal F}(d)$.
For a sequence $d$ of non-negative integers, let ${\cal F}(d)$ be the set of all forests whose degree sequence is $d$. We present closed formulas for $γ_{\max}^{\cal F}(d)=\max\{ γ(F):F\in {\cal F}(d)\}$ and $α_{\min}^{\cal F}(d)=\min\{ α(F):F\in {\cal F}(d)\}$ where $γ(F)$ and $α(F)$ are the domination number and the independence number of a forest $F$, respectively.
We prove that, for every integer $d$ with $d\geq 3$, there is an approximation algorithm for the maximum induced matching problem restricted to $\{ C_3,C_5\}$-free $d$-regular graphs with performance ratio $0.708\bar{3}d+0.425$, which answers a question posed by Dabrowski et al. (Theor. Comput. Sci. 478 (2013) 33-40). Furthermore, we show that every graph with $m$ edges that is $k$-degenerate and of maximum degree at most $d$ with $k<d$, has an induced matching with at least $m/((3k-1)d-k(k+1)+1)$ edges.
A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only if $|S(u)\cap S(v)|\geq p$. We show that every minimal forbidden induced subgraph for the hereditary class ${\cal G}(d,p)$ of graphs whose $p$-intersection number is at most $d$, has order at most $3\cdot 2^{d+1}+1$, and that the exponential dependence on $d$ in this upper bound is necessary. For $p\in \{ d-1,d-2\}$, we provide more explicit results characterizing the graphs in ${\cal G}(d,p)$ without isolated/universal vertices using forbidden induced subgraphs.
The independence number $α(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$α(H)\geq \sum_{u\in V(H)}f_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\geq 2$, where \begin{eqnarray*} f_r(0)&=&1\mbox{, and }\\ f_r(d)&=&\frac{1+\Big((r-1)d^2-d\Big)f_r(d-1)}{1+(r-1)d^2}\mbox{ for $d\geq 1$.} \end{eqnarray*}
A matching $M$ in a graph $G$ is uniquely restricted if there is no matching $M'$ in $G$ that is distinct from $M$ but covers the same vertices as $M$. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs in which some maximum matching is uniquely restricted. Solving a problem posed by Levit and Mandrescu, we characterize the graphs in which every maximum matching is uniquely restricted. Both our characterizations lead to efficient recognition algorithms for the corresponding graphs.
For every positive integer $k$, we show that every graph of order $n$ at least $3k$ with more than $$\max\{{2k-1\choose 2}+(2k-1)(n-(2k-1)),{3k-1\choose 2}+(n-(3k-1))\}$$ edges has $k$ vertex disjoint cycles, which is a best possible density version of a theorem of Corrádi and Hajnal.
We show that a cubic graph $G$ of order $n$ has an induced $2$-regular subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no induced cycle of length more than $4$, and $n>6$, and c) $\left(\frac{1}{4}+ε\right)n$, if the independence number of $G$ is at most $\left(\frac{3}{8}-ε\right)n$. To show the second result we give a precise structural description of cubic $4$-chordal graphs.
We characterize the graphs for which the independence number equals the packing number. As a consequence we obtain simple structural descriptions of the graphs for which (i) the distance-$k$-packing number equals the distance-$2k$-packing number, and (ii) the distance-$k$-matching number equals the distance-$2k$-matching number. This last result considerably simplifies and extends previous results of Cameron and Walker (The graphs with maximum induced matching and maximum matching the same size, Discrete Math. 299 (2005) 49-55). For positive integers $k_1$ and $k_2$ with $k_1<k_2$ and $\lceil(3k_2+1)/2\rceil\leq 2k_1+1$, we prove that it is NP-hard to determine for a given graph whether its distance-$k_1$-packing number equals its distance-$k_2$-packing number.
The Gallai graph $Γ(G)$ of a graph $G$ has the edges of $G$ as its vertices and two distinct vertices $e$ and $f$ of $Γ(G)$ are adjacent in $Γ(G)$ if the edges $e$ and $f$ of $G$ are adjacent in $G$ but do not span a triangle in $G$. Clearly, $Γ(G)$ is a subgraph of the line graph of $G$. While line graphs can be recognized efficiently the complexity of recognizing Gallai graphs is unknown. In the present paper we characterize those graphs whose Gallai graphs are forests or trees, respectively.
If $G$ is a claw-free hamiltonian graph of order $n$ and maximum degree $Δ$ with $Δ\geq 24$, then $G$ has cycles of at least $\min\left\{ n,\left\lceil\frac{3}{2}Δ\right\rceil\right\}-2$ many different lengths.
We prove that a cubic graph with $m$ edges has an induced matching with at least $m/9$ edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and Müller (Induced matchings in subcubic planar graphs, SIAM J. Discrete Math. 26 (2012) 1383-1411) and solves a conjecture of Henning and Rautenbach (Induced matchings in subcubic graphs without short cycles, to appear in Discrete Math.).
Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister \emph{et al.} the argument of which contains a gap. Furthermore, we prove upper bounds on lpt(G) for planar graphs and graphs of bounded tree-width.
Let G=(V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v in V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v in V, there is exactly one d in D dominating v. An edge set M is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph. We give a unified framework for investigating the complexity of these problems on various classes of graphs. In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs. We extend the two problems to hypergraphs and show that ED remains NP-complete on alpha-acyclic hypergraphs, and is solvable in polynomial time on hypertrees, while EED is polynomial on alpha-acyclic hypergraphs and NP-complete on hypertrees.
We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.
Given a sequence A of 2n real numbers, the Even-Rank-Sum problem asks for the sum of the n values that are at the even positions in the sorted order of the elements in A. We prove that, in the algebraic computation-tree model, this problem has time complexity Θ(n log n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008.