Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of $P_4$-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [21] and computer-aided branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and Applications (DMAA

Hamiltonian chordal graphs are not cycle extendible

In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle extendible; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on $n$ vertices for any $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a non-extendible cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendibility in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.Comment: Some results from Section 3 were incorrect and have been removed. To appear in SIAM Journal on Discrete Mathematic

Forbidden minors characterization of partial 3-trees

AbstractA k-tree is formed from a k-complete graph by recursively adding a vertex adjacent to all vertices in an existing k-complete subgraph. The many applications of partial k-trees (subgraphs of k-trees) have motivated their study from both the algorithmic and theoretical points of view. In this paper we characterize the class of partial 3-trees by its set of four minimal forbidden minors (H is a minor of G if H can be obtained from G by a finite sequence of edge-extraction and edge-contradiction operations.

Injective colorings of sparse graphs

Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $\chi_i(G)\leq\Delta(G) + 1$; and if $mad(G) < 42/19$, then $\chi_i(G)=\Delta(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $\Delta(G)\geq 4$. We prove that if $g(G)\geq 9$, then $\chi_i(G)\leq\Delta(G)+1$; similarly, if $g(G)\geq 13$, then $\chi_i(G)=\Delta(G)$.Comment: 10 page

On a connection between the switching separability of a graph and that of its subgraphs

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: if removing any one or two vertices of a graph always results in a switching separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order greater than 4, there is a graph that is not switching separable, but removing any vertex always results in a switching separable subgraph. We show a connection with similar facts on the separability of Boolean functions and reducibility of $n$-ary quasigroups. Keywords: two-graph, reducibility, separability, graph switching, Seidel switching, graph connectivity, $n$-ary quasigroupComment: english: 9 pages; russian: 9 page

Minimal reducible bounds for induced-hereditary properties

AbstractLet (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice Ma we denote by BM(P). Analogously, the set of all minimal reducible bounds of a property P∈La in La is denoted by BL(P).We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice La. Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in La and k-degenerate graphs in Ma

On prime inductive classes of graphs

AbstractLet H[G1,…,Gn] denote a graph formed from unlabelled graphs G1,…,Gn and a labelled graph H=({v1,…,vn},E) replacing every vertex vi of H by the graph Gi and joining the vertices of Gi with all the vertices of those of Gj whenever {vi,vj}∈E(H). For unlabelled graphs G1,…,Gn,H, let φH(G1,…,Gn) stand for the class of all graphs H[G1,…,Gn] taken over all possible orderings of V(H).A prime inductive class of graphs, I(B,C), is said to be a set of all graphs, which can be produced by recursive applying of φH(G1,…,G∣V(H)∣) where H is a graph from a fixed set C of prime graphs and G1,…,G∣V(H)∣ are either graphs from the set B of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, k-trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) [1] Drgas-Burchardt et al. (2010) [6] and Hajós (1961) [10]).This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class P it presents a method for the construction of maximal induced hereditary graph classes contained in P and substitution closed.The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is givenThere is also offered an algebraic view on the class of all prime inductive classes of graphs of the type I({K1},C)