On a unique tree representation for P4-extendible graphs

AbstractSeveral practical applications in computer science and computational linguistics suggest the study of graphs that are unlikely to have more than a few induced paths of length three. These applications have motivated the notion of a cograph, defined by the very strong restriction that no vertex may belong to an induced path of length three. The class of P4-extendible graphs that we introduce in this paper relaxes this restriction, and in fact properly contains the class of cographs, while still featuring the remarkable property of admitting a unique tree representation. Just as in the case of cographs, the class of P4-extendible graphs finds applications to clustering, scheduling, and memory management in a computer system. We give several characterizations for P4-extendible graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-extendible graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

On some special classes of contact B0B_0-VPG graphs

A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact $B_0$-VPG graph if it is a $B_0$-VPG graph admitting a representation with no two paths crossing and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within four special graph classes: chordal graphs, tree-cographs, $P_4$-tidy graphs and $P_5$-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact $B_0$-VPG graphs.Comment: 34 pages, 15 figure

On the p-Connectedness of Graphs – a Survey

A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs. This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions

Representation of graphs by OBDDs

AbstractRecently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P4s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements

Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure

On P_4-tidy graphs

We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15]

The Clique Operator On Graphs With Few P4's

The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. The iterated clique graphs Kn(G) are defined by K0(G)=G and Ki(G)=K(Ki-1(G)),i>0 and K is the clique operator. In this article we use the modular decomposition technique to characterize the K-behaviour of some classes of graphs with few P4's . These characterizations lead to polynomial time algorithms for deciding the K-convergence or K-divergence of any graph in the class. © 2005 Elsevier B.V. All rights reserved.1543485492Babel, L., Kloks, T., Kratochvíl, J., Kratsch, D., Müller, H., Olariu, S., Efficient algorithms for graphs with few P4's (2001) Discrete Math., 235, pp. 29-51Bandelt, H.-J., Prisner, E., Clique graphs and Helly graphs (1991) J. Combin. Theory Ser. B, 51, pp. 34-45Bornstein, C.F., Szwarcfiter, J.L., On clique convergent graphs (1995) Graphs Combin., 11, pp. 213-220Cheen, B.-L., Lih, K.-W., Diameters of iterated clique graphs of chordal graphs (1990) J. Graph Theory, 14, pp. 391-396Cournier, A., Habib, M., A new linear algorithms for modular decomposition (1994) Lecture Notes in Computer Science, 787, pp. 68-84. , Springer, BerlinEscalante, F., Über iterierte clique-graphen (1973) Abh. Math. Sem. Univ. Hamburg, 39, pp. 59-68Frias, M.E., Neumann-Lara, V., Pizaña, M., Dismantlings and iterated clique graphs (2004) Discrete Math., 282, pp. 263-265Giakoumakis, V., Vanherpe, J.-M., On extended P4P-reducible and extended P4-sparse graphs (1997) Theoret. Comput. Sci., 180, pp. 269-286Giakoumakis, V., Roussel, F., Thuillier, H., On P4-tidy graphs (1997) Discrete Math. Theoret. Comput. Sci., 1, pp. 17-41Hamelink, R., A partial characterization of clique graphs (1968) J. Combin. Theory, 5, pp. 192-197Hoang, C., (1985), Doctoral Dissertation, McGill University Montreal, QuebecJamison, B., Olariu, S., A new class of brittle graphs (1989) Stud. Appl. Math., 81, pp. 89-92Jamison, B., Olariu, S., P4-reducible graphs, a class of uniquely tree representable graphs (1989) Stud. Appl. Math., 81, pp. 79-87Jamison, B., Olariu, S., On a unique tree representation for P4-extendible graphs (1991) Discrete Appl. Math., 34, pp. 151-164Jamison, B., Olariu, S., A tree representation for P4-sparse graphs (1992) Discrete Appl. Math., 35, pp. 115-129Larrión, F., De Mello, C.P., Morgana, A., Neumann-Lara, V., Pizaña, M., The clique operator on cographs and serial graphs (2004) Discrete Math., 282, pp. 183-191Larrión, F., Neumann-Lara, V., Clique divergent graphs with unbounded sequence of diameter (1999) Discrete Math., 197-198, pp. 491-501Larrión, F., Neumann-Lara, V., Locally C6 graphs are clique divergent (2000) Discrete Math., 215, pp. 159-170Larrión, F., Neumann-Lara, V., On clique divergent graphs with linear growth (2001) Discrete Math., 245, pp. 139-153Larrión, F., Neumann-Lara, V., Pizaña, M., Whitney triangulations, local girth and iterated clique graphs (2002) Discrete Math., 258, pp. 123-135McConnell, R.M., Spinrad, J.P., Modular decomposition and transitive orientation (1999) Discrete Math., 201, pp. 189-206Neumann-Lara, V., On clique-divergent graphs (1978) Colloq. Internat. CNRS, 260, pp. 313-315. , Problèmes Combinatoires et Théorie des GraphesNeumann-Lara, V., Clique-divergence in graphs (1978) Coll. Math. Soc. Janos Bolyai, 25, pp. 563-569. , Algebraic Methods in Graph Theory, Szeged, Húngary North-Holland, AmsterdamNeumann-Lara, V., Theory of Clique Expansive Graphs, , unpublished manuscriptPizaña, M.A., The icosahedron is clique divergent (2003) Discrete Math., 262, pp. 229-239Prisner, E., Convergence of iterated clique graphs (1992) Discrete Math., 103, pp. 199-207Prisner, E., Hereditary Helly graphs (1993) J. Combin. Math. Combin. Comput., 14, pp. 216-220Prisner, E., Graph dynamics (1995) Pitman Research Notes in Mathematics, 338. , Longman, New YorkSzwarcfiter, J.L., Recognizing clique-Helly graphs (1997) Ars. Combin., 45, pp. 29-32Szwarcfiter, J.L., A survey on clique graphs (2003) Recent Advances in Algorithms and Combinatorics, CMS Books in Mathematics, pp. 109-136. , C. Linhares B. Reed Springer Berli