Characterizing and recognizing probe block graphs (original) (raw)
Related papers
Good characterizations and linear time recognition for 2-probe block graphs
Discrete Applied Mathematics, 2017
Block graphs are graphs in which every block (biconnected component) is a clique. A graph G = (V, E) is said to be an (unpartitioned) k-probe block graph if there exist k independent sets N i ⊆ V , 1 ≤ i ≤ k, such that the graph G ′ obtained from G by adding certain edges between vertices inside the sets N i , 1 ≤ i ≤ k, is a block graph; if the independent sets N i are given, G is called a partitioned k-probe block graph. In this paper we give good characterizations for 2-probe block graphs, in both unpartitioned and partitioned cases. As an algorithmic implication, partitioned and unpartitioned probe block graphs can be recognized in linear time, improving a recognition algorithm of cubic time complexity previously obtained by Chang et al. [Block-graph width, Theoretical Computer Science 412 (2011), 2496-2502].
Characterizing Block Graphs in Terms of One-vertex Extensions
Block graphs has been extensively studied for many decades. In this paper we present a new characterization of the class in terms of one-vertex extensions. To this purpose, a specific representation based on the concept of boundary cliques is presented, bringing about some properties of the graph.
Partitioned probe comparability graphs
Theoretical Computer Science, 2008
Given a class of graphs G, a graph G is a probe graph of G if its vertices can be partitioned into a set of probes and an independent set of nonprobes such that G can be embedded into a graph of G by adding edges between certain nonprobes. If the partition of the vertices is part of the input, we call G a partitioned probe graph of G. In this paper we show that there exists a polynomial-time algorithm for the recognition of partitioned probe graphs of comparability graphs. This immediately leads to a polynomial-time algorithm for the recognition of partitioned probe graphs of cocomparability graphs. We then show that a partitioned graph G is a partitioned probe permutation graph if and only if G is at the same time a partitioned probe graph of comparability and cocomparability graphs.
The complexity of clique graph recognition
Theoretical Computer Science, 2009
Consider finite, simple and undirected graphs. V and E denote the vertex set and the edge set of the graph G, respectively. A complete set of G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. The clique family of G is denoted by C(G). The clique graph of G is the intersection graph of C(G). The clique operator, K, assigns to each graph G its clique graph which is denoted by K(G). On the other hand, say that G is a clique graph if G belongs to the image of the clique operator, i.e. if there exists a graph H such that G = K (H). Clique operator and its image were widely studied. First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I<n, for some positive integer n; and the behavior under these operators of parameters such as the number of vertices or diameter were studied in [5,8,9,12,26,30] and more recently in [7,14,21-23, 29], For several classes of graphs, the image of the class under the clique operator was characterized [10,18,19,24,34,37]; and, in some cases, also the inverse image of the class [16,28,35], Results of the previous bibliography can be found in the survey [39], Clique graphs have been much studied as intersection graphs and are included in several books [11,25,33], In this paper we are concerned with the time complexity of the problem of recognizing clique graphs, this is the time complexity of the following decision problem.
Partitioning a graph into two pieces, each isomorphic to the other or to its complement
Discrete Mathematics, 2005
A simple graph G has the generalized-neighbour-closed-co-neighbour property, or is a gncc graph, if for all vertices x of G, the subgraph, induced by the set of neighbours of x, is isomorphic to the subgraph, induced by the set of non-neighbours of x, or is isomorphic to its complement. If every vertex x satisfies the first condition (that is, the subgraphs, induced by its set of neighbours, and by its set of non-neighbours, are isomorphic), then the graph has the neighbour-closed-co-neighbour property, or is an ncc graph. In [A. Bonato, R. Nowakowski, Partitioning a graph into two isomorphic pieces, J. Graph Theory, 44 (2003) 1-14], the ncc graphs were characterized and a polynomial time algorithm was given for their recognition. In this paper we show that all gncc graphs are also ncc, that is, we prove that the two families of graphs, defined above, are identical. Finally, we present some of the properties of an interesting family of graphs, that is derived from the proof of the claim above, and we give a polynomial time algorithm to recognize such graphs.
A characterisation of clique-width through nested partitions
Discrete Applied Mathematics, 2015
Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, preserve the tree structures and that they are computable in polynomial time. We apply our characterisation to linear clique-width. And we relate our characterisation to a clique-width characterisation by Heule and Szeider that is used to reduce the clique-width decision problem to a satisfiability problem.
Matemática Contemporânea, 2010
A complete set of a graph G is a subset of V G whose elements are pairwise adjacent. A clique is a maximal complete set. The clique graph of G, denoted by K(G), is the intersection graph of the family of cliques of G. The clique graph recognition problem asks whether a given graph is a clique graph. This problem was classified recently as NP-complete after being open for 30 years. The complexity of this decision problem is open for very structured and well studied classes of graphs such as planar graphs and chordal graphs. We propose the study of split clique graphs.
Polynomial-time recognition of clique-width ≤3 graphs
Discrete Applied Mathematics, 2012
Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of cliquewidth at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n 2 m)) to recognize graphs of clique-width at most 3.
On connectivity of the semi-splitting block graph of a graph
Acta Universitatis Sapientiae, Informatica
A graph G is said to be a semi-splitting block graph if there exists a graph H such that SB(H) ≌ G. This paper establishes a characterisation of semi-splitting block graphs based on the partition of the vertex set of G. The vertex (edge) connectivity and p-connectedness (p-edge connectedness) of SB(G) are examined. For all integers a, b with 1 < a < b, the existence of the graph G for which κ (G) = a, κ (SB(G)) = b and λ (G) = a, λ (SB(G)) = b are proved independently. The characterization of graphs with κ(SB(G)) = κ (G) and a necessary condition for graphs with κ (SB(G)) = λ (SB(G)) are achieved.