Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

Irredundant sets, Ramsey numbers, multicolor Ramsey numbers

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in a blue subgraph or a $n$-element independent set in a red subgraph. The \emph{multicolor irredundant Ramsey number} $s(t_{1},\ldots,t_{l})$ is the minimum $r$ such that every $l$-coloring of the edges of the complete graph $K_{r}$ on $r$ vertices has a monochromatic irredundant set of size $s_{i}$ for certain $1\leq i\leq l$. Firstly, we improve the upper bound for the mixed Ramsey number $t(3,n)$, and using this result, we verify a special case of a conjecture proposed by Chen, Hattingh, and Rousseau for $m=4$. Secondly, we obtain a new upper bound for $s(3,9)$, and using Krivelevich's method, we establish an asymptotic lower bound for CO-irredundant Ramsey number of $K_{N}$, which extends Krivelevich's result on $s(m,n)$. Thirdly, we prove a lower bound for the multicolor irredundant Ramsey number by a random and probability method which has been used to improve the lower bound for multicolor Ramsey numbers. Finally, we give a lower bound for the irredundant multiplicity.Comment: 23 pages, 1 figur

Lossy gossip and composition of metrics

We study the monoid generated by n-by-n distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension n(n-1)/2, and we compute the structure of this fan for n up to 5. The monoid captures gossip among n gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid, as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.Comment: Minor textual edits, final versio

Asymptotic lower bounds for Gallai-Ramsey functions and numbers

For two graphs $G,H$ and a positive integer $k$, the \emph{Gallai-Ramsey number} ${\rm gr}_k(G,H)$ is defined as the minimum number of vertices $n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $H$. If $G$ and $H$ are both complete graphs, then we call it \emph{Gallai-Ramsey function} ${\rm GR}_k(s,t)$, which is the minimum number of vertices $n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow copy of $K_s$ or a monochromatic copy of $K_t$. In this paper, we derive some lower bounds for Gallai-Ramsey functions and numbers by Lov\'{o}sz Local Lemma.Comment: 11 page

Changing upper irredundance by edge addition

AbstractDenote the upper irredundance number of a graph G by IR(G). A graph G is IR-edge-addition-sensitive if its upper irredundance number changes whenever an edge of Ḡ is added to G. Specifically, G is IR-edge-critical (IR+-edge-critical, respectively) if IR(G+e)<IR(G) (IR(G+e)>IR(G), respectively) for each edge e of Ḡ. We show that if G is IR-edge-addition-sensitive, then G is either IR-edge-critical or IR+-edge-critical. We obtain properties of the latter class of graphs, particularly in the case where β(G)=IR(G)=2 (where β(G) denotes the vertex independence number of G). This leads to an infinite class of IR+-edge-critical graphs where IR(G)=2

On the Parameterized Complexity of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to the parameters tw and (k, c4), where tw and c4 are the treewidth and the number of cycles with length 4 of the input graph. We also prove that the problem is fixed parameter tractable with respect to the parameter k for the line graphs and every proper minor-closed class of graphs (including planar graphs)