Vinícius Fernandes dos Santos

On the Carathéodory number of interval and graph convexities

Inspired by a result of Caratheodory [Uber den Variabilitatsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911) 193-217], the Caratheodory number of a convexity space is defined as the smallest integer k such that for every subset U of the ground set V and every element u in the convex hull of U, there is a subset F of U with at most k elements such that u in the convex hull of F. We study the Caratheodory number for generalized interval convexities and for convexity spaces derived from finite graphs. We establish structural properties, bounds, and hardness results.

Algorithmic and structural aspects of the P 3-Radon number

The generalization of classical results about convex sets in ℝn to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P3-convexity on graphs. P3-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon’s classical convexity result. We establish hardness results and describe efficient algorithms for trees.

An upper bound on the P 3 -Radon number

The generalization of classical results about convex sets in R n to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P 3 -convexity on graphs.A set R of vertices of a graph G is P 3 -convex if no vertex in V ( G ) ? R has two neighbours in R . The P 3 -convex hull of a set of vertices is the smallest P 3 -convex set containing it. The P 3 -Radon number r ( G ) of a graph G is the smallest integer r such that every set R of r vertices of G has a partition R = R 1 ? R 2 such that the P 3 -convex hulls of R 1 and R 2 intersect. We prove that r ( G ) ? 2 3 ( n ( G ) + 1 ) + 1 for every connected graph G and characterize all extremal graphs.

On the radon number for p 3 convexity

The generalization of classical results about convex sets in ℝn to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P3-convexity on graphs. P3-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radons classical convexity result. We establish hardness results, describe efficient algorithms for trees, and prove a best-possible bound on the Radon number of connected graphs.

An upper bound on the P3-Radon number

The generalization of classical results about convex sets in R n to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P 3 -convexity on graphs.A set R of vertices of a graph G is P 3 -convex if no vertex in V ( G ) ? R has two neighbours in R . The P 3 -convex hull of a set of vertices is the smallest P 3 -convex set containing it. The P 3 -Radon number r ( G ) of a graph G is the smallest integer r such that every set R of r vertices of G has a partition R = R 1 ? R 2 such that the P 3 -convex hulls of R 1 and R 2 intersect. We prove that r ( G ) ? 2 3 ( n ( G ) + 1 ) + 1 for every connected graph G and characterize all extremal graphs.

The convexity of induced paths of order three and applications: Complexity aspects

Abstract In this paper, we introduce a new convexity on graphs similar to the well known P 3 -convexity, which we will call P 3 ∗ -convexity. We show that several P 3 ∗ -convexity parameters (hull number, convexity number, Caratheodory number, Radon number, interval number and percolation time) are NP-hard even on bipartite graphs. We prove a strong relationship between this convexity and the well known geodesic convexity, which implies several NP-hardness results for the latter. In order to show that, we prove that the hull number for the P 3 -convexity is NP-hard even for subgraphs of grids and that the convexity number for the P 3 -convexity is NP-hard even for bipartite graphs with diameter 3. We also obtain linear time algorithms to determine those parameters for the above mentioned convexities for cographs and P 4 -sparse graphs.

A strategy for clustering students minimizing the number of bus stops for solving the school bus routing problem

In this work we tackle the bus stop selection step for the School Bus Routing Problem (SBRP). Our goal is to minimize the number of bus stops in order to assign all students to a bus stop respecting a home-to-bus-stop walking distance constraint. Our strategy creates a large number of possible bus stops points in a road network and uses a pseudo-random constructive heuristic algorithm to assign students to a bus stops. Our approach is tested on a real georeferenced data of a Brazilian city and is compared with a different methodology. Results demonstrate that the proposed approach is able to find good solutions for this optimization problem. Besides, the higher the number of possible points to install bus stops, the smaller is the number of bus stops required to attend all students.

Irreversible conversion processes with deadlines

Abstract Given a graph G, a deadline t d ( u ) and a time-dependent threshold f ( u , t ) for every vertex u of G, we study sequences C = ( c 0 , c 1 , … ) of 0/1-labelings c i of the vertices of G such that for every t ∈ N , we have c t ( u ) = 1 if and only if either c t − 1 ( u ) = 1 or at least f ( u , t − 1 ) neighbors v of u satisfy c t − 1 ( v ) = 1 . The sequence C models the spreading of a property/commodity within a network and it is said to converge to 1 on time, if c t d ( u ) ( u ) = 1 for every vertex u of G, that is, if every vertex u has the spreading property/received the spreading good by time t d ( u ) . We study the smallest number irr ( G , t d , f ) of vertices u with initial label c 0 ( u ) equal to 1 that result in a sequence C converging to 1 on time. If G is a forest or a clique, we present efficient algorithms computing irr ( G , t d , f ) . Furthermore, we prove lower and upper bounds relying on counting and probabilistic arguments. For special choices of t d and f, the parameter irr ( G , t d , f ) coincides with well-known graph parameters related to domination and independence in graphs.

On recognition of threshold tolerance graphs and their complements

Abstract A graph G = ( V , E ) is a threshold tolerance graph if each vertex v ∈ V can be assigned a weight w v and a tolerance t v such that two vertices x , y ∈ V are adjacent if w x + w y ≥ min ( t x , t y ) . Currently, the most efficient recognition algorithm for threshold tolerance graphs is the algorithm of Monma, Reed, and Trotter which has an O ( n 4 ) runtime. We give an O ( n 2 ) algorithm for recognizing threshold tolerance and their complements, the co-threshold tolerance (co-TT) graphs, resolving an open question of Golumbic, Weingarten, and Limouzy.

On Minimal and Minimum Hull Sets

Abstract A graph convexity ( G , C ) is a graph G together with a collection C of subsets of V ( G ) , called convex sets, such that ∅ , V ( G ) ∈ C and C is closed under intersections. For a set U ⊆ V ( G ) , the hull of U, denoted H ( U ) , is the smallest convex set containing U. If H ( U ) = V ( G ) , then U is a hull set of G. Motivated by the theory of well covered graphs, which investigates the relation between maximal and maximum independent sets of a graph, we study the relation between minimal and minimum hull sets. We concentrate on the P 3 convexity, where convex sets are closed under adding common neighbors of their elements.