Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTS-Reconfiguration of Dominating Sets in circle and circular-arc graphs
150
0
0.0
(
0
)
Added by
Alice Joffard
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S=<D_1:=D_s,...,D_l:=D_t> of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.
rate research
Read More
In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear algebraic tools, we prove that any strongly connected graph with even period has at least two independent dominating sets, generalizing several of the results of Cary, Cary, and Prabhu. We prove that determining the period of the graph is not sufficient to determine the existence of an independent dominating set by constructing a few examples of infinite families of graphs. We show that the direct analogue of Vizings Conjecture does not hold for independent domination number in directed graphs by providing two infinite families of graphs. We initialize the study of time complexity for independent domination in directed graphs, proving that the existence of an independent dominating set in directed acyclic graphs and strongly connected graphs with even period are in the time complexity class $P$. We also provide an algorithm for determining existence of an independent dominating set for digraphs with period greater than $1$.
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs, which is an important class of geometric intersection graphs. In this paper we give sharp bounds for the hyperbolicity constant of (finite and infinite) circular-arc graphs. Moreover, we obtain bounds for the hyperbolicity constant of the complement and line of any circular-arc graph. In order to do that, we obtain new results about regular, chordal and line graphs which are interesting by themselves.
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA 21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollobas, and Morris [Eur. J. Comb. 06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.
Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $Dsubseteq V$ such that every vertex $uin Vsetminus D$ is adjacent to at least one vertex $vin D$. That is, the set $D$ satisfies the condition that $|N[v]cap D|geq 1$ for each $vin V$, where $N[v]$ is the closed neighborhood of $v$. In this paper, we study two variants of the classical dominating set problem: $boldmath{k}$-tuple dominating set ($k$-DS) problem and Liars dominating set (LDS) problem, and obtain several algorithmic and hardness results. On the algorithmic side, we present a constant factor ($frac{11}{2}$)-approximation algorithm for the Liars dominating set problem on unit disk graphs. Then, we obtain a PTAS for the $boldmath{k}$-tuple dominating set problem on unit disk graphs. On the hardness side, we show a $Omega (n^2)$ bits lower bound for the space complexity of any (randomized) streaming algorithm for Liars dominating set problem as well as for the $boldmath{k}$-tuple dominating set problem. Furthermore, we prove that the Liars dominating set problem on bipartite graphs is W[2]-hard.
For $kgeq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to ${1,2,ldots,k}$ such that $c(u) eq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For a family of graphs ${cal H}$, a graph $G$ is ${cal H}$-free if $G$ does not contain any graph from ${cal H}$ as an induced subgraph. Let $C_s$ be the $s$-vertex cycle. In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of $3$-Colouring for $(C_3,ldots,C_s)$-free graphs and $H$-free graphs where $H$ is some polyad. Here, we prove for certain small values of $s$ that $3$-Colouring is polynomial-time solvable for $C_s$-free graphs of diameter $2$ and $(C_4,C_s)$-free graphs of diameter $2$. In fact, our results hold for the more general problem List $3$-Colouring. We complement these results with some hardness result for diameter $4$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
