Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComputing Subset Transversals in $H$-Free Graphs
162
0
0.0
(
0
)
Added by
Matthew Johnson
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $sgeq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $sgeq 1$.
rate research
Read More
A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NP-hard and study approximation algorithms. We develop theoretical tools that help to study this problem.
The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest $r$ such that $L^{r}(G)$ has a Hamiltonian cycle. Checking if $h(G) = k$ is NP-hard for any fixed integer $k geq 0$ even for subcubic graphs $G$. We study the parameterized complexity of this problem with the parameter treewidth, $tw(G)$, and show that we can find $h(G)$ in time $O*((1 + 2^{(omega + 3)})^{tw(G)})$ where $omega$ is the matrix multiplication exponent and the $O*$ notation hides polynomial factors in input size. The NP-hard Eulerian Steiner Subgraph problem takes as input a graph $G$ and a specified subset $K$ of terminal vertices of $G$ and asks if $G$ has an Eulerian (that is: connected, and with all vertices of even degree.) subgraph $H$ containing all the terminals. A second result (and a key ingredient of our algorithm for finding $h(G)$) in this work is an algorithm which solves Eulerian Steiner Subgraph in $O*((1 + 2^{(omega + 3)})^{tw(G)})$ time.
Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.
For every constant $d geq 3$ and $epsilon > 0$, we give a deterministic $mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Theta(n)$ vertices that is $epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2sqrt{d-1} + epsilon$ (excluding the single trivial eigenvalue of~$d$).
A graph $G = (V,E)$ is a double-threshold graph if there exist a vertex-weight function $w colon V to mathbb{R}$ and two real numbers $mathtt{lb}, mathtt{ub} in mathbb{R}$ such that $uv in E$ if and only if $mathtt{lb} le mathtt{w}(u) + mathtt{w}(v) le mathtt{ub}$. In the literature, those graphs are studied as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs, which gives connections to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [COCOON 2018] ran in $O(n^6)$ time, where $n$ is the number of vertices.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
