arXiv subjects
Publications and source records attributed to Felix Joos.
The intersection graphs of stars in some tree are known as substar graphs. In this paper we give a characterization of substar graphs by the list of minimal forbidden induced subgraphs. This corrects a flaw in the main result of Chang, Jacobson, Monma and West (Subtree and substar intersection numbers, Discrete Appl. Math. 44, 205-220 (1993)) and this leads to a different list of minimal forbidden induced subgraphs.
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 boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.
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.
As observed by Rautenbach and Sereni (arXiv:1302.5503) there is a gap in the proof of the theorem of Balister et al. (Longest paths in circular arc graphs, Combin. Probab. Comput., 13, No. 3, 311-317 (2004)), which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.
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.).
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$ vertices, $G^n$ the $n$-th Cartesian power of $G$, $α_i$ be the number of vertices of degree $i$ of $G$, $λ$ be a positive real number, and $G^n_p$ be the graph obtained from $G^n$ by deleting every edge independently with probability $1-p$. If $\sum_{i}α_i(1-p)^i=λ^{\frac{1}{n}}$, then $\lim_{n\rightarrow \infty}\mathbb{P}[G^n_p {\rm\ is\ connected}]=\exp(-λ)$. This result extends known results for regular graphs. The main result implies that the threshold probability does not depend on the graph structure of $G$ itself, but only on the degree sequence of the graph.
We prove that $\frac{7}{4}α(G)+β(G)\geq n(G)$ and $α(G)+\frac{3}{2}β(G)\geq n(G)$ for every triangle-free graph $G$ with maximum degree at most $4$, where $α(G)$ is the independence number and $β(G)$ is the matching number of $G$, respectively. These results are sharp for a graph on $13$ vertices. Furthermore we show $χ(G)\leq \frac{7}{4}ω(G)$ for $\{3K_1,K_1\cup K_5\}$-free graphs, where $χ(G)$ is the chromatic number and $ω(G)$ is the clique number of $G$, respectively.