Strong resolvability in product graphs.

En aquesta tesi s'estudia la dimensió mètrica forta de grafs producte. Els resultats més importants de la tesi se centren en la recerca de relacions entre la dimensió mètrica forta de grafs producte i la dels seus factors, juntament amb altres invariants d'aquests factors. Així, s'han estudiat els següents productes de grafs: producte cartesià, producte directe, producte fort, producte lexicogràfic, producte corona, grafs unió, suma cartesiana, i producte arrel, d'ara endavant "grafs producte". Hem obtingut fórmules tancades per la dimensió mètrica forta de diverses famílies no trivials de grafs producte que inclouen, per exemple, grafs bipartits, grafs vèrtexs transitius, grafs hamiltonians, arbres, cicles, grafs complets, etc, i hem donat fites inferiors i superiors generals, expressades en termes d'invariants dels grafs factors, com ara, l'ordre, el nombre d'independència, el nombre de cobriment de vèrtexs, el nombre d'aparellament, la connectivitat algebraica, el nombre de cliqué, i el nombre de cliqué lliure de bessons. També hem descrit algunes classes de grafs producte, on s'assoleixen aquestes fites. És conegut que el problema de trobar la dimensió mètrica forta d'un graf connex es pot transformar en el problema de trobar el nombre de cobriment de vèrtexs de la seva corresponent graf de resolubilitat forta. En aquesta tesi hem aprofitat aquesta eina i hem trobat diverses relacions entre el graf de resolubilitat forta de grafs producte i els grafs de resolubilitat forta dels seus factors. Per exemple, és notable destacar que el graf de resolubilitat forta del producte cartesià de dos grafs és isomorf al producte directe dels grafs de resolubilitat forta dels seus factors.En esta tesis se estudia la dimensión métrica fuerte de grafos producto. Los resultados más importantes de la tesis se centran en la búsqueda de relaciones entre la dimensión métrica fuerte de grafos producto y la de sus factores, junto con otros invariantes de estos factores. Así, se han estudiado los siguientes productos de grafos: producto cartesiano, producto directo, producto fuerte, producto lexicográfico, producto corona, grafos unión, suma cartesiana, y producto raíz, de ahora en adelante "grafos producto". Hemos obtenido fórmulas cerradas para la dimensión métrica fuerte de varias familias no triviales de grafos producto que incluyen, por ejemplo, grafos bipartitos, grafos vértices transitivos, grafos hamiltonianos, árboles, ciclos, grafos completos, etc, y hemos dado cotas inferiores y superiores generales, expresándolas en términos de invariantes de los grafos factores, como por ejemplo, el orden, el número de independencia, el número de cubrimiento de vértices, el número de emparejamiento, la conectividad algebraica, el número de cliqué, y el número de cliqué libre de gemelos. También hemos descrito algunas clases de grafos producto, donde se alcanzan estas cotas. Es conocido que el problema de encontrar la dimensión métrica fuerte de un grafo conexo se puede transformar en el problema de encontrar el número de cubrimiento de vértices de su correspondiente grafo de resolubilidad fuerte. En esta tesis hemos aprovechado esta herramienta y hemos encontrado varias relaciones entre el grafo de resolubilidad fuerte de grafos producto y los grafos de resolubilidad fuerte de sus factores. Por ejemplo, es notable destacar que el grafo de resolubilidad fuerte del producto cartesiano de dos grafos es isomorfo al producto directo de los grafos de resolubilidad fuerte de sus factores.In this thesis we study the strong metric dimension of product graphs. The central results of the thesis are focused on finding relationships between the strong metric dimension of product graphs and that of its factors together with other invariants of these factors. We have studied the following products: Cartesian product graphs, direct product graphs, strong product graphs, lexicographic product graphs, corona product graphs, join graphs, Cartesian sum graphs, and rooted product graphs, from now on ``product graphs''. We have obtained closed formulaes for the strong metric dimension of several nontrivial families of product graphs involving, for instance, bipartite graphs, vertex-transitive graphs, Hamiltonian graphs, trees, cycles, complete graphs, etc., or we have given general lower and upper bounds, and have expressed these in terms of invariants of the factor graphs like, for example, order, independence number, vertex cover number, matching number, algebraic connectivity, clique number, and twin-free clique number. We have also described some classes of product graphs where these bounds are achieved. It is known that the problem of finding the strong metric dimension of a connected graph can be transformed to the problem of finding the vertex cover number of its strong resolving graph. In the thesis we have strongly exploited this tool. We have found several relationships between the strong resolving graph of product graphs and that of its factor graphs. For instance, it is remarkable that the strong resolving graph of the Cartesian product of two graphs is isomorphic to the direct product of the strong resolving graphs of its factors

A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure

Fixing number of co-noraml product of graphs

An automorphism of a graph $G$ is a bijective mapping from the vertex set of $G$ to itself which preserves the adjacency and the non-adjacency relations of the vertices of $G$. A fixing set $F$ of a graph $G$ is a set of those vertices of $G$ which when assigned distinct labels removes all the automorphisms of $G$, except the trivial one. The fixing number of a graph $G$, denoted by $fix(G)$, is the smallest cardinality of a fixing set of $G$. The co-normal product $G_1\ast G_2$ of two graphs $G_1$ and $G_2$, is a graph having the vertex set $V(G_1)\times V(G_2)$ and two distinct vertices $(g_1, g_2), (\acute{g_1}, \acute{g_2})$ are adjacent if $g_1$ is adjacent to $\acute{g_1}$ in $G_1$ or $g_2$ is adjacent to $\acute{g_2}$ in $G_2$. We define a general co-normal product of $k\geq 2$ graphs which is a natural generalization of the co-normal product of two graphs. In this paper, we discuss automorphisms of the co-normal product of graphs using the automorphisms of its factors and prove results on the cardinality of the automorphism group of the co-normal product of graphs. We prove that $max\{fix(G_1), fix(G_2)\}\leq fix(G_1\ast G_2)$, for any two graphs $G_1$ and $G_2$. We also compute the fixing number of the co-normal product of some families of graphs.Comment: 13 page

Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure

On the limiting distribution of the metric dimension for random forests

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure

3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page