The clique number of an undirected graph G is the maximum order of a complete subgraph of G and i... more The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well-known lower bound for the chromatic number of G. Every proper k-coloring of G may be viewed as a homomorphism (an edge-preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this paper, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all k ≥ 4.
Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs ... more Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs to be assigned exactly one of the colors from a given palette of c colors. At an instant of time only the facilities of one particular color are 'active' and all other facilities are 'dormant'. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over R and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of n and c. We propose an optimal coloring scheme for the number of facilities n being a multiple of c as well as for the general case where n is not a multiple of c. When n is a multiple of c, we compute an optimal scheme in Θ(n) time. For the general case, we propose a coloring scheme that returns the optimal in O(n 2 log n) time.
We solve a movement problem in which there are n sensors in path network in plane, where any sens... more We solve a movement problem in which there are n sensors in path network in plane, where any sensor communicates only with its two immediate neighbors and only at a given maximum communication distance λ. Initially, some of the inter-sensor distances may be more than λ. We need to move sensors so that each sensor is in the communication range of its two neighbors, keeping the path topology intact. The problem is to minimize the maximum movement of any sensor. We present an O(n 3)-time algorithm to compute the new positions of sensors to establish transmission connectivity, called λ-connectivity in the paper, in a convex path-network which minimizes the maximum movement among the sensors. We also generalize our algorithm for ring, non-convex path, tethered and heterogeneous networks.
In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n g... more In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in $$\mathfrak {R}^2$$ with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to $$\mathfrak {R}^3$$ by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in $$\mathfrak {R}^d$$ where center of the minimum enclosing ball lies inside a given ball. We show that similar problem of minimum intersecting/stabbing ball for set of hyper planes in $$\mathfrak {R}^d$$ can also be solved using similar techniques. We also show how minimum intersecting disk with center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time complexity for fixed dimension.
In this paper we present algorithms for computing 1-center of a set of points for convex polyhedr... more In this paper we present algorithms for computing 1-center of a set of points for convex polyhedral distance function in \(\mathfrak {R}^d\) for any d. Given polyhedral P of size m, the running time of our algorithm for computing 1-center of n points in \(\mathfrak {R}^2\) for convex polygonal distance function \(d_P\) is \(O(nm\log ^2 m)\). For \(d>2\), we present an \(O(3^{3d^2} nm^2\log ^d m)\) algorithm to compute 1-center of n points in \(\mathfrak {R}^d\) for convex polyhedral distance function \(d_P\), \(|P|=m\). Both the algorithms are linear time for fixed d and fixed polyhedron P.
We solve a movement problem in which there are n sensors in path network in plane, where any sens... more We solve a movement problem in which there are n sensors in path network in plane, where any sensor communicates only with its two immediate neighbors and only at a given maximum communication distance λ. Initially, some of the inter-sensor distances may be more than λ. We need to move sensors so that each sensor is in the communication range of its two neighbors, keeping the path topology intact. The problem is to minimize the maximum movement of any sensor. We present an O(n 3)-time algorithm to compute the new positions of sensors to establish transmission connectivity, called λ-connectivity in the paper, in a convex path-network which minimizes the maximum movement among the sensors. We also generalize our algorithm for ring, non-convex path, tethered and heterogeneous networks.
Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs ... more Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs to be assigned exactly one of the colors from a given palette of c colors. At an instant of time only the facilities of one particular color are 'active' and all other facilities are 'dormant'. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over R and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of n and c. We propose an optimal coloring scheme for the number of facilities n being a multiple of c as well as for the general case where n is not a multiple of c. When n is a multiple of c, we compute an optimal scheme in Θ(n) time. For the general case, we propose a coloring scheme that returns the optimal in O(n 2 log n) time.
HAL (Le Centre pour la Communication Scientifique Directe), 2022
A proper n-coloring of a graph G is an assignment of colors from {1,. .. , n} to its vertices suc... more A proper n-coloring of a graph G is an assignment of colors from {1,. .. , n} to its vertices such that no two adjacent vertices get assigned the same color. The chromatic number of G, denoted by χ(G), refers to the smallest n such that G admits a proper n-coloring. This notion naturally extends to edge-colorings (resp. total-colorings) when edges (resp. both vertices and edges) are to be colored, and this provides other parameters of G: its chromatic index χ (G) and its total chromatic number χ (G). These coloring notions are among the most fundamental ones of the graph coloring theory. As such, they gave birth to hundreds of studies dedicated to several of their aspects, including generalizations to more general structures such as oriented graphs. They include notably the notions of oriented n-colorings and oriented n-arc-colorings, which stand as natural extensions of their undirected counterparts, and which have been receiving increasing attention. Our goal is to introduce a missing piece in this line of work, namely the oriented counterparts of proper n-total-colorings and total chromatic number. We first define these notions and show that they share properties and connections with oriented (arc) colorings that are reminiscent of those shared by their undirected counterparts. We then focus on understanding the oriented total chromatic number of particular types of oriented graphs, such as oriented forests, cycles, and some planar graphs. Finally, we establish a full complexity dichotomy for the problem of determining whether an oriented graph is totally k-colorable. Throughout this work, each of our results is compared to what is known regarding the oriented chromatic number and oriented chromatic index. We also disseminate some directions for further research on the oriented total chromatic number.
Abstract In this paper, we first present a linear-time algorithm to find the smallest circle encl... more Abstract In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in ℜ 2 with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to ℜ 3 by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in ℜ d where the center of the minimum enclosing ball lies inside a given ball. We show that similar problem of computing minimum intersecting/stabbing ball for a set of hyper planes in ℜ d can also be solved using similar techniques. We also show how the minimum intersecting disk with the center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time in input size for fixed dimension.
The clique number of an undirected graph G is the maximum order of a complete subgraph of G and i... more The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well-known lower bound for the chromatic number of G. Every proper k-coloring of G may be viewed as a homomorphism (an edge-preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this paper, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all k ≥ 4.
Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs ... more Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs to be assigned exactly one of the colors from a given palette of c colors. At an instant of time only the facilities of one particular color are 'active' and all other facilities are 'dormant'. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over R and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of n and c. We propose an optimal coloring scheme for the number of facilities n being a multiple of c as well as for the general case where n is not a multiple of c. When n is a multiple of c, we compute an optimal scheme in Θ(n) time. For the general case, we propose a coloring scheme that returns the optimal in O(n 2 log n) time.
We solve a movement problem in which there are n sensors in path network in plane, where any sens... more We solve a movement problem in which there are n sensors in path network in plane, where any sensor communicates only with its two immediate neighbors and only at a given maximum communication distance λ. Initially, some of the inter-sensor distances may be more than λ. We need to move sensors so that each sensor is in the communication range of its two neighbors, keeping the path topology intact. The problem is to minimize the maximum movement of any sensor. We present an O(n 3)-time algorithm to compute the new positions of sensors to establish transmission connectivity, called λ-connectivity in the paper, in a convex path-network which minimizes the maximum movement among the sensors. We also generalize our algorithm for ring, non-convex path, tethered and heterogeneous networks.
In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n g... more In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in $$\mathfrak {R}^2$$ with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to $$\mathfrak {R}^3$$ by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in $$\mathfrak {R}^d$$ where center of the minimum enclosing ball lies inside a given ball. We show that similar problem of minimum intersecting/stabbing ball for set of hyper planes in $$\mathfrak {R}^d$$ can also be solved using similar techniques. We also show how minimum intersecting disk with center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time complexity for fixed dimension.
In this paper we present algorithms for computing 1-center of a set of points for convex polyhedr... more In this paper we present algorithms for computing 1-center of a set of points for convex polyhedral distance function in \(\mathfrak {R}^d\) for any d. Given polyhedral P of size m, the running time of our algorithm for computing 1-center of n points in \(\mathfrak {R}^2\) for convex polygonal distance function \(d_P\) is \(O(nm\log ^2 m)\). For \(d>2\), we present an \(O(3^{3d^2} nm^2\log ^d m)\) algorithm to compute 1-center of n points in \(\mathfrak {R}^d\) for convex polyhedral distance function \(d_P\), \(|P|=m\). Both the algorithms are linear time for fixed d and fixed polyhedron P.
We solve a movement problem in which there are n sensors in path network in plane, where any sens... more We solve a movement problem in which there are n sensors in path network in plane, where any sensor communicates only with its two immediate neighbors and only at a given maximum communication distance λ. Initially, some of the inter-sensor distances may be more than λ. We need to move sensors so that each sensor is in the communication range of its two neighbors, keeping the path topology intact. The problem is to minimize the maximum movement of any sensor. We present an O(n 3)-time algorithm to compute the new positions of sensors to establish transmission connectivity, called λ-connectivity in the paper, in a convex path-network which minimizes the maximum movement among the sensors. We also generalize our algorithm for ring, non-convex path, tethered and heterogeneous networks.
Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs ... more Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs to be assigned exactly one of the colors from a given palette of c colors. At an instant of time only the facilities of one particular color are 'active' and all other facilities are 'dormant'. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over R and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of n and c. We propose an optimal coloring scheme for the number of facilities n being a multiple of c as well as for the general case where n is not a multiple of c. When n is a multiple of c, we compute an optimal scheme in Θ(n) time. For the general case, we propose a coloring scheme that returns the optimal in O(n 2 log n) time.
HAL (Le Centre pour la Communication Scientifique Directe), 2022
A proper n-coloring of a graph G is an assignment of colors from {1,. .. , n} to its vertices suc... more A proper n-coloring of a graph G is an assignment of colors from {1,. .. , n} to its vertices such that no two adjacent vertices get assigned the same color. The chromatic number of G, denoted by χ(G), refers to the smallest n such that G admits a proper n-coloring. This notion naturally extends to edge-colorings (resp. total-colorings) when edges (resp. both vertices and edges) are to be colored, and this provides other parameters of G: its chromatic index χ (G) and its total chromatic number χ (G). These coloring notions are among the most fundamental ones of the graph coloring theory. As such, they gave birth to hundreds of studies dedicated to several of their aspects, including generalizations to more general structures such as oriented graphs. They include notably the notions of oriented n-colorings and oriented n-arc-colorings, which stand as natural extensions of their undirected counterparts, and which have been receiving increasing attention. Our goal is to introduce a missing piece in this line of work, namely the oriented counterparts of proper n-total-colorings and total chromatic number. We first define these notions and show that they share properties and connections with oriented (arc) colorings that are reminiscent of those shared by their undirected counterparts. We then focus on understanding the oriented total chromatic number of particular types of oriented graphs, such as oriented forests, cycles, and some planar graphs. Finally, we establish a full complexity dichotomy for the problem of determining whether an oriented graph is totally k-colorable. Throughout this work, each of our results is compared to what is known regarding the oriented chromatic number and oriented chromatic index. We also disseminate some directions for further research on the oriented total chromatic number.
Abstract In this paper, we first present a linear-time algorithm to find the smallest circle encl... more Abstract In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in ℜ 2 with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to ℜ 3 by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in ℜ d where the center of the minimum enclosing ball lies inside a given ball. We show that similar problem of computing minimum intersecting/stabbing ball for a set of hyper planes in ℜ d can also be solved using similar techniques. We also show how the minimum intersecting disk with the center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time in input size for fixed dimension.
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