Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRepresenting graphs as the intersection of cographs and threshold graphs
87
0
0.0
(
0
)
Added by
Mathew Francis
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
A graph $G$ is said to be the intersection of graphs $G_1,G_2,ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=cdots=V(G_k)$ and $E(G)=E(G_1)cap E(G_2)capcdotscap E(G_k)$. For a graph $G$, $mathrm{dim}_{COG}(G)$ (resp. $mathrm{dim}_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $mathrm{dim}_{COG}(G)leqmathrm{tw}(G)+2$, (b) $mathrm{dim}_{TH}(G)leqmathrm{pw}(G)+1$, and (c) $mathrm{dim}_{TH}(G)leqchi(G)cdotmathrm{box}(G)$, where $mathrm{tw}(G)$, $mathrm{pw}(G)$, $chi(G)$ and $mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $mathrm{dim}_{COG}(G)$ and $mathrm{dim}_{TH}(G)$ when $G$ belongs to some special graph classes.
rate research
Read More
Bir{o} et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. We negatively answer the 25-year-old question of Bir{o} et al. which asks if $H$-graphs can be recognized in polynomial time, for a fixed graph $H$. We prove that it is NP-complete if $H$ contains the diamond graph as a minor. We provide a polynomial-time algorithm recognizing $T$-graphs, for each fixed tree $T$. When $T$ is a star $S_d$ of degree $d$, we have an $O(n^{3.5})$-time algorithm. We give FPT- and XP-time algorithms solving the minimum dominating set problem on $S_d$-graphs and $H$-graphs parametrized by $d$ and the size of $H$, respectively. The algorithm for $H$-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on $H$-graphs. If $H$ contains the double-triangle as a minor, we prove that $H$-graphs are GI-complete and that the clique problem is APX-hard. The clique problem can be solved in polynomial time if $H$ is a cactus graph. When a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. We show that both the $k$-clique and the list $k$-coloring problems are solvable in FPT-time on $H$-graphs (parameterized by $k$ and the treewidth of $H$). In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that $H$-graphs have at most $n^{O(|H|)}$ minimal separators which allows us to apply the meta-algorithmic framework of Fomin et al. (2015) to show that for each fixed $t$, finding a maximum induced subgraph of treewidth $t$ can be done in polynomial time. When $H$ is a cactus, we improve the bound to $O(|H|n^2)$.
For any class $mathcal{C}$ of bipartite graphs, we define quasi-$cal C$ to be the class of all graphs $G$ such that every bipartition of $G$ belongs to $cal C$. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.
Given two independent sets $I, J$ of a graph $G$, and imagine that a token (coin) is placed at each vertex of $I$. The Sliding Token problem asks if one could transform $I$ to $J$ via a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors so that the resulting set of vertices where tokens are placed remains independent. This problem is $mathsf{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded-treewidth. In this paper, we show that Sliding Token can be solved efficiently for cactus graphs and block graphs, and give upper bounds on the length of a transformation sequence between any two independent sets of these graph classes. Our algorithms are designed based on two main observations. First, all structures that forbid the existence of a sequence of token slidings between $I$ and $J$, if exist, can be found in polynomial time. A sufficient condition for determining no-instances can be easily derived using this characterization. Second, without such forbidden structures, a sequence of token slidings between $I$ and $J$ does exist. In this case, one can indeed transform $I$ to $J$ (and vice versa) using a polynomial number of token-slides.
Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its toric ideal $P_G$ is a complete intersection or not. Whenever $P_G$ is a complete intersection, the algorithm also returns a minimal set of generators of $P_G$. Moreover, we prove that if $G$ is a connected graph and $P_G$ is a complete intersection, then there exist two induced subgraphs $R$ and $C$ of $G$ such that the vertex set $V(G)$ of $G$ is the disjoint union of $V(R)$ and $V(C)$, where $R$ is a bipartite ring graph and $C$ is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if $R$ is $2$-connected and $C$ is connected, we list the families of graphs whose toric ideals are complete intersection.
The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves well in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
