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Register a new userThinness of product graphs
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Added by
Flavia Bonomo
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${rm dim}(X)$ of a comparability graph $X$ is the dimension of any transitive orientation of X, and by $k$-DIM we denote the class of comparability graphs $X$ with ${rm dim}(X) le k$. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordans characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For $k ge 4$, we show that every finite group can be realized as the automorphism group of some graph in $k$-DIM, and testing graph isomorphism for $k$-DIM is GI-complete.
Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any $(d + 2)$-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for $d = 1$. Bousquet and Perarnau proved that a linear transformation exists for between $(2d + 2)$-colorings. It is open to determine if this bound can be reduced. In this note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist.
The objective of the well-known Towers of Hanoi puzzle is to move a set of disks one at a time from one of a set of pegs to another, while keeping the disks sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs.
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