Avances en Matemática Discreta en Andalucía Volumen 4, 2015
El lóbulo óptico de la Drosophila (mosca de la fruta) es una estructura altamente sofisticada con... more El lóbulo óptico de la Drosophila (mosca de la fruta) es una estructura altamente sofisticada con más de 60.000 neuronas de más de 70 tipos diferentes que permite realizar una serie de tareas altamente complejas como el procesamiento del color, la detección de movimiento o el seguimiento de cambios ambientales. En un cerebro tan pequeño alcanzar estas tareas requiere un alto nivel organizativo y estructurado con diferentes tipos neuronales especializados en funciones específicas. Para analizar el complejo linaje celular del lóbulo óptico se ha generado una gran cantidad de clones y se han modelizado las relaciones usando técnicas de detección de comunidades en grafos ponderados.
Let E X (ν; {C 3 ,. .. , C n }) denote the set of graphs G of order ν that contain no cycles of l... more Let E X (ν; {C 3 ,. .. , C n }) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n + 1 cycle? We prove that the diameter of G is at most n − 1, and present several results concerning the above question: the girth of G is g = n + 1 if (i) ν ≥ n + 5, diameter equal to n − 1 and minimum degree at least 3; (ii) ν ≥ 12, ν ∈ {15, 80, 170} and n = 6. Moreover, if ν = 15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν ≥ 2n − 3 and n ≥ 7 the girth is at most 2n − 5. We also show that the answer to the question is negative for ν ≤ n + 1 + (n − 2)/2 .
For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integ... more For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. C., et al. ex(n, T K p ) which represents the maximum number of edges of a graph on n vertices free of a topological minor T K p of a complete graph on p vertices (see Bollobás' excellent monograph devoted to this subject and the contributions on this topic ). The second was stated by Zarankiewicz [10] who studied the maximum size of a bipartite graph on (m, n) vertices, denoted by z(m, n; s, t) that contains no bipartite complete K (s,t) subgraph with s vertices in the m-class and t vertices in the n-class. For a survey of this problem we also refer the reader to Section VI.2 of the book by Bollobás [3]. Most of the contributions are bounds for the function z(m, n; s, t) when s, t are fixed and m, n are much larger than s, t [11-13]. Other contributions provide exact values of the extremal function [14-16]. Recent results on some problems involving the existence of a complete bipartite graph or a subdivision of a complete bipartite graph as a subgraph can be found in the literature [17-20]. Böhme et al. [21] studied the size of a k-connected graph free of either an induced path of a given length or a subdivision of a complete bipartite graph. Kühn and Osthus [22] proved that for any graph H and for every integer s there exists a function f = f (H, s) such that every graph of size at least f contains either a K s,s as a subgraph or an induced subdivision of H. Meyer [17] also relates the size of a graph with the property of containing a minor of K s,t .
Applicable Analysis and Discrete Mathematics, 2015
Maximal connectivity and superconnectivity in a network are two important features of its reliabi... more Maximal connectivity and superconnectivity in a network are two important features of its reliability. In this paper, using graph terminology, we first give a lower bound for the vertex connectivity of the strong product of two networks and then we prove that the resulting structure is more reliable than its generators. Namely, sufficient conditions for a strong product of two networks to be maximally connected and superconnected are given
Journal of Optimization Theory and Applications, 2007
Protection, surveillance or other types of coverage services of mobile points call for different,... more Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey's movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpredictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.
The optic lobes of the fruit fly Drosophila melanogaster form a highly wired neural network compo... more The optic lobes of the fruit fly Drosophila melanogaster form a highly wired neural network composed of roughly 130.000 neurons of more than 80 different types. How neuronal diversity arises from very few cell progenitors is a central question in developmental neurobiology. We use the optic lobe of the fruit fly as a paradigm to understand how neuroblasts, the neural stem cells, generate multiple neuron types. Although the development of the fly brain has been the subject of extensive research, very little is known about the lineage relationships of the cell types forming the adult optic lobes. Here we perform a large-scale lineage bioinformatics analysis using the graph theory. We generated a large collection of cell clones that genetically label the progeny of neuroblasts and built a database to draw graphs showing the lineage relationships between cell types. By establishing biological criteria that measures the strength of the neuronal relationships and applying community detection tools we have identified eight clusters of neurons. Each cluster contains different cell types that we pose are the product of eight distinct classes of neuroblasts. Three of these clusters match the available lineage data, supporting the predictive value of the analysis. Finally, we show that the neuronal progeny of a neuroblast do not have preferential innervation patterns, but instead become part of different layers and neuropils. Here we establish a new methodology that helps understanding the logic of Drosophila brain development and can be applied to the more complex vertebrate brains.
International Journal of Applied Mathematics and Computer Science, 2016
This paper extends to infinite graphs the most general extremal issues, which are problems of det... more This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.
ABSTRACT The authors study the maximum number of edges in a graph of order n not containing the c... more ABSTRACT The authors study the maximum number of edges in a graph of order n not containing the complete bipartite graph K t,t as a subgraph. The upper bound in terms of n and t together with corresponding sharpness examples is presented.
A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated ver... more A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edgeconnectivity λ (G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ-optimal if λ (G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defined as ξ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. The P-diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D 1 (resp. D 2) be the P-diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is λ-optimal if D 1 g − 2 and D 2 g − 5. For even girth we obtain a similar result. Second, let F ⊂ V (G) with |F | = δ − 1, δ 2, being the minimum degree of G. Using the property Q of being vertices of G − F we prove that a graph with girth g / ∈ {4, 6, 8} is λ-optimal if this Q-diameter is at most 2 (g − 3)/2 .
The extremal number ex(n; MK p) denotes the maximum number of edges of a graph of order n contain... more The extremal number ex(n; MK p) denotes the maximum number of edges of a graph of order n containing no complete graph K p as a minor. In this paper we give the exact value of the extremal number ex(n; MK p) for (5n + 9)/8 p (2n − 1)/3 provided that n − p 24. Indeed we show that this number is the size of the Turán Graph T 2p−n−1 (n) and this graph is the only extremal graph.
The restricted connectivity (G) of a connected graph G is defined as the minimum cardinality of a... more The restricted connectivity (G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex u has all its neighbors in X; the superconnectivity 1 (G) is defined similarly, this time considering only vertices u in G − X, hence 1 (G) (G). The minimum edge-degree of G is (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding 1 (G) (G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17-31] and guaranteeing 1 (G) = (G) = (G) under some additional constraints.
Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph ... more Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h) such that g is odd and h g + 3 is even has high (vertex-)connectivity if its diameter is at most h − 3. The edge version of all results is also studied.
The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large ... more The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters (and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m , G p of minimum degrees (G m), (G p), respectively, we show that (G m * G p) is lower bounded by both (G m) + (G p) and (G p) + (G m), an improvement of what is known for the edge-connectivity of G m × G p .
ABSTRACT For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and ... more ABSTRACT For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. KeywordsBipartite graphs–extremal graph theory–topological minor
ABSTRACT Consider the extremal problem of finding the maximum number of edges of a graph of order... more ABSTRACT Consider the extremal problem of finding the maximum number of edges of a graph of order n not containing the complete graph K p as a minor. The authors solve the problem for a certain range of values of p and n. A new upper bound for the problem is also established.
Given a bipartite graph G with m and n vertices, respectively, in its vertices classes, and given... more Given a bipartite graph G with m and n vertices, respectively, in its vertices classes, and given two integers s, t such that 2 ≤ s ≤ t, 0 ≤ m−s ≤ n−t, and m+n ≤ 2s+t−1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.
Avances en Matemática Discreta en Andalucía Volumen 4, 2015
El lóbulo óptico de la Drosophila (mosca de la fruta) es una estructura altamente sofisticada con... more El lóbulo óptico de la Drosophila (mosca de la fruta) es una estructura altamente sofisticada con más de 60.000 neuronas de más de 70 tipos diferentes que permite realizar una serie de tareas altamente complejas como el procesamiento del color, la detección de movimiento o el seguimiento de cambios ambientales. En un cerebro tan pequeño alcanzar estas tareas requiere un alto nivel organizativo y estructurado con diferentes tipos neuronales especializados en funciones específicas. Para analizar el complejo linaje celular del lóbulo óptico se ha generado una gran cantidad de clones y se han modelizado las relaciones usando técnicas de detección de comunidades en grafos ponderados.
Let E X (ν; {C 3 ,. .. , C n }) denote the set of graphs G of order ν that contain no cycles of l... more Let E X (ν; {C 3 ,. .. , C n }) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n + 1 cycle? We prove that the diameter of G is at most n − 1, and present several results concerning the above question: the girth of G is g = n + 1 if (i) ν ≥ n + 5, diameter equal to n − 1 and minimum degree at least 3; (ii) ν ≥ 12, ν ∈ {15, 80, 170} and n = 6. Moreover, if ν = 15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν ≥ 2n − 3 and n ≥ 7 the girth is at most 2n − 5. We also show that the answer to the question is negative for ν ≤ n + 1 + (n − 2)/2 .
For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integ... more For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. C., et al. ex(n, T K p ) which represents the maximum number of edges of a graph on n vertices free of a topological minor T K p of a complete graph on p vertices (see Bollobás' excellent monograph devoted to this subject and the contributions on this topic ). The second was stated by Zarankiewicz [10] who studied the maximum size of a bipartite graph on (m, n) vertices, denoted by z(m, n; s, t) that contains no bipartite complete K (s,t) subgraph with s vertices in the m-class and t vertices in the n-class. For a survey of this problem we also refer the reader to Section VI.2 of the book by Bollobás [3]. Most of the contributions are bounds for the function z(m, n; s, t) when s, t are fixed and m, n are much larger than s, t [11-13]. Other contributions provide exact values of the extremal function [14-16]. Recent results on some problems involving the existence of a complete bipartite graph or a subdivision of a complete bipartite graph as a subgraph can be found in the literature [17-20]. Böhme et al. [21] studied the size of a k-connected graph free of either an induced path of a given length or a subdivision of a complete bipartite graph. Kühn and Osthus [22] proved that for any graph H and for every integer s there exists a function f = f (H, s) such that every graph of size at least f contains either a K s,s as a subgraph or an induced subdivision of H. Meyer [17] also relates the size of a graph with the property of containing a minor of K s,t .
Applicable Analysis and Discrete Mathematics, 2015
Maximal connectivity and superconnectivity in a network are two important features of its reliabi... more Maximal connectivity and superconnectivity in a network are two important features of its reliability. In this paper, using graph terminology, we first give a lower bound for the vertex connectivity of the strong product of two networks and then we prove that the resulting structure is more reliable than its generators. Namely, sufficient conditions for a strong product of two networks to be maximally connected and superconnected are given
Journal of Optimization Theory and Applications, 2007
Protection, surveillance or other types of coverage services of mobile points call for different,... more Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey's movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpredictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.
The optic lobes of the fruit fly Drosophila melanogaster form a highly wired neural network compo... more The optic lobes of the fruit fly Drosophila melanogaster form a highly wired neural network composed of roughly 130.000 neurons of more than 80 different types. How neuronal diversity arises from very few cell progenitors is a central question in developmental neurobiology. We use the optic lobe of the fruit fly as a paradigm to understand how neuroblasts, the neural stem cells, generate multiple neuron types. Although the development of the fly brain has been the subject of extensive research, very little is known about the lineage relationships of the cell types forming the adult optic lobes. Here we perform a large-scale lineage bioinformatics analysis using the graph theory. We generated a large collection of cell clones that genetically label the progeny of neuroblasts and built a database to draw graphs showing the lineage relationships between cell types. By establishing biological criteria that measures the strength of the neuronal relationships and applying community detection tools we have identified eight clusters of neurons. Each cluster contains different cell types that we pose are the product of eight distinct classes of neuroblasts. Three of these clusters match the available lineage data, supporting the predictive value of the analysis. Finally, we show that the neuronal progeny of a neuroblast do not have preferential innervation patterns, but instead become part of different layers and neuropils. Here we establish a new methodology that helps understanding the logic of Drosophila brain development and can be applied to the more complex vertebrate brains.
International Journal of Applied Mathematics and Computer Science, 2016
This paper extends to infinite graphs the most general extremal issues, which are problems of det... more This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.
ABSTRACT The authors study the maximum number of edges in a graph of order n not containing the c... more ABSTRACT The authors study the maximum number of edges in a graph of order n not containing the complete bipartite graph K t,t as a subgraph. The upper bound in terms of n and t together with corresponding sharpness examples is presented.
A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated ver... more A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edgeconnectivity λ (G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ-optimal if λ (G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defined as ξ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. The P-diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D 1 (resp. D 2) be the P-diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is λ-optimal if D 1 g − 2 and D 2 g − 5. For even girth we obtain a similar result. Second, let F ⊂ V (G) with |F | = δ − 1, δ 2, being the minimum degree of G. Using the property Q of being vertices of G − F we prove that a graph with girth g / ∈ {4, 6, 8} is λ-optimal if this Q-diameter is at most 2 (g − 3)/2 .
The extremal number ex(n; MK p) denotes the maximum number of edges of a graph of order n contain... more The extremal number ex(n; MK p) denotes the maximum number of edges of a graph of order n containing no complete graph K p as a minor. In this paper we give the exact value of the extremal number ex(n; MK p) for (5n + 9)/8 p (2n − 1)/3 provided that n − p 24. Indeed we show that this number is the size of the Turán Graph T 2p−n−1 (n) and this graph is the only extremal graph.
The restricted connectivity (G) of a connected graph G is defined as the minimum cardinality of a... more The restricted connectivity (G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex u has all its neighbors in X; the superconnectivity 1 (G) is defined similarly, this time considering only vertices u in G − X, hence 1 (G) (G). The minimum edge-degree of G is (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding 1 (G) (G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17-31] and guaranteeing 1 (G) = (G) = (G) under some additional constraints.
Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph ... more Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h) such that g is odd and h g + 3 is even has high (vertex-)connectivity if its diameter is at most h − 3. The edge version of all results is also studied.
The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large ... more The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters (and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m , G p of minimum degrees (G m), (G p), respectively, we show that (G m * G p) is lower bounded by both (G m) + (G p) and (G p) + (G m), an improvement of what is known for the edge-connectivity of G m × G p .
ABSTRACT For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and ... more ABSTRACT For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. KeywordsBipartite graphs–extremal graph theory–topological minor
ABSTRACT Consider the extremal problem of finding the maximum number of edges of a graph of order... more ABSTRACT Consider the extremal problem of finding the maximum number of edges of a graph of order n not containing the complete graph K p as a minor. The authors solve the problem for a certain range of values of p and n. A new upper bound for the problem is also established.
Given a bipartite graph G with m and n vertices, respectively, in its vertices classes, and given... more Given a bipartite graph G with m and n vertices, respectively, in its vertices classes, and given two integers s, t such that 2 ≤ s ≤ t, 0 ≤ m−s ≤ n−t, and m+n ≤ 2s+t−1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K (s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.
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