Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page

Alliance Partitions in Graphs.

For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong alliance partition number) is the maximum cardinality of a partition of V into defensive alliances (respectively, strong defensive alliances). The global (strong) alliance partition number is defined similarly. For each parameter we give both general bounds and exact values. Our major results include exact values for the alliance partition number of grid graphs and for the global alliance partition number of caterpillars

Very Cost Effective Partitions in Graphs

For a graph G=(V,E) and a set of vertices S, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V -S than in S. A bipartition pi={S, V- S} is called very cost effective if both S and V- S are very cost effective sets. Not all graphs have a very cost effective bipartition, for example, the complete graphs of odd order do not. We consider several families of graphs G, including Cartesian products and cacti graphs, to determine whether G has a very cost effective bipartition

Global Secure Sets Of Trees And Grid-like Graphs

Let G = (V, E) be a graph and let S ⊆ V be a subset of vertices. The set S is a defensive alliance if for all x ∈ S, |N[x] ∩ S| ≥ |N[x] − S|. The concept of defensive alliances was introduced in [KHH04], primarily for the modeling of nations in times of war, where allied nations are in mutual agreement to join forces if any one of them is attacked. For a vertex x in a defensive alliance, the number of neighbors of x inside the alliance, plus the vertex x, is at least the number of neighbors of x outside the alliance. In a graph model, the vertices of a graph represent nations and the edges represent country boundaries. Thus, if the nation corresponding to a vertex x is attacked by its neighbors outside the alliance, the attack can be thwarted by x with the assistance of its neighbors in the alliance. In a different subject matter, [FLG00] applies graph theory to model the world wide web, where vertices represent websites and edges represent links between websites. A web community is a subset of vertices of the web graph, such that every vertex in the community has at least as many neighbors in the set as it has outside. So, a web community C satisfies ∀x ∈ C, |N[x] ∩ C| \u3e |N[x] − C|. These sets are very similar to defensive alliances. They are known as strong defensive alliances in the literature of alliances in graphs. Other areas of application for alliances and related topics include classification, data clustering, ecology, business and social networks. iii Consider the application of modeling nations in times of war introduced in the first paragraph. In a defensive alliance, any attack on a single member of the alliance can be successfully defended. However, as will be demonstrated in Chapter 1, a defensive alliance may not be able to properly defend itself when multiple members are under attack at the same time. The concept of secure sets is introduced in [BDH07] for exactly this purpose. The non-empty set S is a secure set if every subset X ⊆ S, with the assistance of vertices in S, can successfully defend against simultaneous attacks coming from vertices outside of S. The exact definition of simultaneous attacks and how such attacks may be defended will be provided in Chapter 1. In [BDH07], the authors presented an interesting characterization for secure sets which resembles the definition of defensive alliances. A non-empty set S is a secure set if and only if ∀X ⊆ S, |N[X] ∩ S| ≥ |N[X] − S| ([BDH07], Theorem 11). The cardinality of a minimum secure set is the security number of G, denoted s(G). A secure set S is a global secure set if it further satisfies N[S] = V . The cardinality of a minimum global secure set of G is the global security number of G, denoted γs(G). In this work, we present results on secure sets and global secure sets. In particular, we treat the computational complexity of finding the security number of a graph, present algorithms and bounds for the global security numbers of trees, and present the exact values of the global security numbers of paths, cycles and their Cartesian products

Alliance polynomial and hyperbolicity in regular graphs

One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by the use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial of a graph. The alliance polynomial of a graph G with order n and maximum degree δ_1 is the polynomial A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) 〗, where A{_k}(G) is the number of exact defensive k-alliances in G. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: • Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. • Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. • Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected ∆-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ∆-regular. If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ >= 0 : X is δ-hyperbolic }. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is the hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K_4 or to K_3,3, the inequalities 5k/4 <= δ (G ̅) <=3k/2 hold, if k is the length of every edge in G. --------------------Uno de los problemas abiertos en la teoría de grafos es la caracterización de cualquier grafo por un polinomio. La investigación en este área ha sido impulsada en gran parte por las ventajas que ofrece el uso de las computadoras que hacen que trabajar con grafos sea más simple. En esta Tesis introducimos el polinomio de alianza de un grafo. El polinomio de alianza de un grafo G con orden n y grado m´aximo δ_1 es el polinomio A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) , donde A{_k}(G) es el n´umero de k alianzas defensivas exactas en G. También desarrollamos e implementamos un algoritmo que calcula de manera eficiente el polinomio de alianza. En este trabajo obtenemos algunas propiedades de A(G; x) y sus coeficientes para: • Grafos caminos, ciclos, completos y estrellas. En particular, hemos demostrado que se caracterizan mediante sus polinomios de alianza. • Grafos cúbicos (grafos con todos sus vértices de grado 3), ya que son una clase muy interesante de grafos con muchas aplicaciones. Hemos demostrado que sus polinomios de alianza verifican unimodalidad. Además, calculamos el polinomio de alianza para grafos cúbicos de orden pequeño, los cuales satisfacen unicidad. • Grafos regulares (grafos con todos sus vértices de igual grado). En particular, se caracteriza el grado de los grafos regulares por el n´umero de coeficientes distintos de cero de su polinomio de alianza. Además, se demuestra que la familia de polinomios de alianza de grafos conexos _-regulares con grado pequeño es muy especial, ya que no contiene polinomios de alianza de grafos conexos que no sean _-regulares. Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] and [x3x1] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de todo triángulo geodésico T de X está contenido en la δ-vecindad de la unóon de los otros dos lados. Se denota por δ(X) la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ > 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Hemos obtenido información acerca de la constante de hiperbolicidad de los grafos cúbicos; dichos grafos son muy importantes en el estudio de la hiperbolicidad, ya que para cualquier grafo G con grado máximo acotado existe un grafo cúbico G∗ tal que G es hiperbólico si y sólo si G∗ es hiperbólico. En esta memoria conseguimos caracterizar los grafos cúbicos con constante de hiperbolicidad pequeña. Además, se obtienen cotas para la constante de hiperbolicidad del grafo complemento de un grafo cúbico; nuestro principal resultado dice que para cualquier grafo cúbico finito G no isomorfo a K4 o K3,3, se cumple la relación 5k/4 <= δ (G ̅) <=3k/2, donde k es la longitud de todas las aristas en G.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván; Secretario: Eva Tourís Lojo; Vocal: Sergio Bermudo Navarret

Domination on hyperbolic graphs

If k ≥ 1 and G = (V, E) is a finite connected graph, S ⊆ V is said a distance k-dominating set if every vertex v ∈ V is within distance k from some vertex of S. The distance k-domination number γ kw (G) is the minimum cardinality among all distance k-dominating sets of G. A set S ⊆ V is a total dominating set if every vertex v ∈ V satisfies δS (v) ≥ 1 and the total domination number, denoted by γt(G), is the minimum cardinality among all total dominating sets of G. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ(G) and some domination parameters of a graph G. The results in this work are inequalities, such as γkw(G) ≥ 2δ(G)/(2k + 1) and δ(G) ≤ γt(G)/2 + 3.Supported by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain, and a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain