Minimum Number Of Arcs Research Articles

On computing optimal temporal branchings and spanning subgraphs

We extend the concept of out/in-branchings spanning the vertices of a digraph to temporal graphs, which are digraphs where arcs are available only at prescribed times. While the literature has focused on minimum weight/earliest arrival time Temporal Out-Branchings (tob), we solve the problem for other optimization criteria (travel duration, departure time, number of transfers, total waiting time, traveling time). For some criteria we provide a log linear algorithm for computing such branchings, while for others we prove that deciding the existence of a spanning tob is NP-complete. The same results hold for optimal temporal in-branchings. We also investigate the related problem of computing a spanning temporal subgraph with the minimum number of arcs and optimizing a chosen criterion; this problem turns out to be always NP-hard. The hardness results are quite surprising, as computing optimal paths between nodes is always polynomial-time.

On the minimum number of arcs in 4‐dicritical oriented graphs

AbstractThe dichromatic number of a digraph is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph is ‐dicritical if and each proper subdigraph of satisfies . For integers and , we define (resp., ) as the minimum number of arcs possible in a ‐dicritical digraph (resp., oriented graph). Kostochka and Stiebitz have shown that . They also conjectured that there is a constant such that for and large enough. This conjecture is known to be true for . In this work, we prove that every 4‐dicritical oriented graph on vertices has at least arcs, showing the conjecture for . We also characterise exactly the 4‐dicritical digraphs on vertices with exactly arcs.

Various Bounds on the Minimum Number of Arcs in a $k$-Dicritical Digraph

The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. 
 We prove various bounds on the minimum number of arcs in a $k$-dicritical digraph, a structural result on $k$-dicritical digraphs and a result on list-dicolouring. We characterise $3$-dicritical digraphs $G$ with $(k-1)|V(G)| + 1$ arcs. For $k \geq 4$, we characterise $k$-dicritical digraphs $G$ on at least $k+1$ vertices and with $(k-1)|V(G)| + k-3$ arcs, generalising a result of Dirac. We prove that, for $k \geq 5$, every $k$-dicritical digraph $G$ has at least $(k-\frac 1 2 - \frac 1 {k-1}) |V(G)| - k(\frac 1 2 - \frac 1 {k-1})$ arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree $2(k-1)$ of a $k$-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.

Spectral Slater index of tournaments

The Slater index $i(T)$ of a tournament $T$ is the minimum number of arcs that must be reversed to make $T$ transitive. In this paper, we define a parameter $\Lambda(T)$ from the spectrum of the skew-adjacency matrix of $T$, called the spectral Slater index. This parameter is a measure of remoteness between the spectrum of $T$ and that of a transitive tournament. We show that $\Lambda(T)\leq8\, i(T)$ and we characterize the tournaments with maximal spectral Slater index. As an application, an improved lower bound on the Slater index of doubly regular tournaments is given.

On the Italian reinforcement number of a digraph

The Italian reinforcement number of a digraph is the minimum number of arcs that have to be added to the digraph in order to decrease the Italian domination number. In this paper, we present some new sharp upper bounds on the Italian reinforcement number of a digraph. We also determine the exact values of the Italian reinforcement number of the Cartesian products of directed paths and directed cycles: $ P_2\square P_n $, $ P_3\square P_n $, $ P_3\square C_n $, $ C_3\square P_n $ and $ C_3\square C_n $.

Hull and Geodetic Numbers for Some Classes of Oriented Graphs

An oriented graph D is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from u to v with minimum number of arcs in D is an (u, v)-geodesic, for every u, v ∈ V(D). A set S ⊆ V(D) is (geodesically) convex if, for every u, v ∈ S, all the vertices in each (u, v)-geodesic and in each (v, u)-geodesic are in S. For every S ⊆ V(D) the (convex) hull of S is the smallest convex set containing S and it is denoted by [S]. A hull set of D is a set S ⊆ V(D) whose hull is V(D). The cardinality of a minimum hull set is the hull number of D and it is denoted by hn→(D). A geodetic set of D is a set S ⊆ V(D) such that each vertex of D lies in an (u, v)-geodesic, for some u, v ∈ S. The cardinality of a minimum geodetic set is the geodetic number of D and it is denoted by gn→(D).In this work, we first present an upper bound for the hull number of oriented split graphs. Then, we turn our attention to the computational complexity of determining such parameters. We first show that computing hn→(D) is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing gn→(D) is also NP-hard for directed acyclic graphs (DAG). Finally, we present a positive result by showing how to compute such parameters in polynomial time when the input graph is an oriented cactus.

Rainbow reinforcement numbers in digraphs

Let [Formula: see text] be a finite and simple digraph. A [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) of a digraph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the set of in-neighbors of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a digraph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a [Formula: see text]RDF of [Formula: see text]. The [Formula: see text]-rainbow reinforcement number [Formula: see text] of a digraph [Formula: see text] is the minimum number of arcs that must be added to [Formula: see text] in order to decrease the [Formula: see text]-rainbow domination number. In this paper, we initiate the study of [Formula: see text]-rainbow reinforcement number in digraphs and we present some sharp bounds for [Formula: see text]. In particular, we determine the [Formula: see text]-rainbow reinforcement number of some classes of digraphs.

A Two-Criterion Lexicographic Algorithm for Finding All Shortest Paths in Networks

An algorithm for finding all shortest paths in an undirected network is considered. The following two criteria are used: the minimum number of arcs in a path and a minimum path length. The algorithm is analyzed for complexity, and it is empirically shown that, with increasing the network density, its computational efficiency becomes higher than that of the Floyd algorithm adequately modified to find the shortest path with the use of a two-step criterion.

Strongly connected orientations of plane graphs

We study the problem of orienting a subset of edges of a given plane graph such that the resulting sub-digraph is strongly connected and spans all vertices of the graph. We are interested in orientations with minimum number of arcs which at the same time produce a digraph with smallest possible stretch factor. Such orientations have applications into the problem of establishing strongly connected sensor network when sensors are equipped with directional antennae.We present three constructions for such orientations. Let G=(V,E) be a 2-edge connected plane graph and let Φ(G) be the degree of the largest face in G. Our constructions are based on a face coloring of G, say with λ colors.The first construction gives a strongly connected orientation with at most (2−4λ−6λ(λ−1))|E| arcs and the stretch factor at most Φ(G)−1. The second construction gives a strongly connected orientation with |E| arcs and the stretch factor at most (Φ(G)−1)⌈λ+12⌉. The third construction can be applied to plane graphs which are 3-edge connected. It uses a particular 6-face coloring and for any integer k≥1 it produces a strongly connected orientation with at most (1−k10(k+1))|E| arcs and the stretch factor at most Φ2(G)(Φ(G)−1)2k+4. Since the stretch factor solely depends only on Φ(G), λ, and k, if these three parameters are bounded, our constructions result in orientations with bounded stretch factor.

OPTIMIZATION OF TREE-STRUCTURED GAS DISTRIBUTION NETWORK USING ANT COLONY OPTIMIZATION: A CASE STUDY

An Ant Colony Optimization (ACO) algorithm is proposed for optimal tree-structured natural gas distribution network. Design of pipelines, facilities, and equipment systems are necessary tasks to configure an optimal natural gas network. A mixed integer programming model is formulated to minimize the total cost in the network. The aim is to optimize pipe diameter sizes so that the location-allocation cost is minimized. Pipeline systems in natural gas network must be designed based on gas flow rate, length of pipe, gas maximum drop pressure allowance, and gas maximum velocity allowance. We use the information regarding gas flow rates and pipe diameter sizes considering gas pressure and velocity restrictions. We apply the Minimum Spanning Tree (MST) technique to obtain a network with minimum number of arcs, spanning all the nodes with no cycle. As a main contribution here, we present and use an ant colony optimization algorithm for solving the problem. The proposed method is applied to a real life situation. Our obtained results are compared with the ones obtained by an exact method. The results show that ACO is an effective approach for gas distribution network optimization. A case study in Mazandaran gas company in Iran is conducted to illustrate the validity and effectiveness of the proposed approach.

DIAMETER VULNERABILITY OF DIRECTED CYCLES AND DIRECTED TORI

The diameter of a graph is important for the network as it measures the maximum communication delay between any pair of processors. In this paper, we study the diameter vulnerability of the directed cycle [Formula: see text] and the directed torus [Formula: see text] with respect to arc addition, that is, what is the minimum number of arcs the addition of which reduces the diameter by at least k.

Optimal pipe diameter sizing in a tree-structured gas network: a case study

We design an optimal pipe diameter sizing in a tree-structured natural gas network. Design of pipeline, facility and equipment systems are necessary tasks to configure an optimal natural gas network. A mixed-integer programming model is formulated to minimise the total cost in the network. The aim is to optimise pipe diameter sizes so that the location-allocation cost is minimised. Pipeline systems in natural gas network must be designed based on gas flow rate, length of pipe, gas maximum drop pressure allowance and gas maximum velocity allowance. We use information based on relationship among gas flow rates and pipe diameter sizes considering gas pressure and velocity restrictions. We apply the minimum spanning tree technique to obtain a network with minimum number of arcs, no cycles and all the spanned nodes. A case study in Mazandaran Gas Company in Iran is conducted to illustrate the validity of the proposed model.

Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs

In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648–662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.

The bondage number in complete t-partite digraphs

The bondage number of a digraph D is the minimum number of arcs whose removal results in a new digraph with larger domination number. In this paper, we determine the bondage number in complete t-partite digraph K → ( n 1 , n 2 , … , n t ) .

Primitive digraphs with smallest large exponent

A primitive digraph D on n vertices has large exponent if its exponent, γ ( D ) , satisfies α n ⩽ γ ( D ) ⩽ w n , where α n = ⌊ w n / 2 ⌋ + 2 and w n = ( n - 1 ) 2 + 1 . It is shown that the minimum number of arcs in a primitive digraph D on n ⩾ 5 vertices with exponent equal to α n is either n + 1 or n + 2 . Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α n and n + 2 arcs. These constructions extend to digraphs with some exponents between α n and w n . A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α n and n + 1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n + 1 or n + 2 .

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Given a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ+ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor $c<10\sqrt{5}-21\approx1.36$ the minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Given a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ+ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor $c<10\sqrt{5}-21\approx1.36$ the minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.

Segmentation and leaf sequencing for intensity modulated arc therapy

A common method in generating intensity modulated radiation therapy (IMRT) plans consists of a three step process: an optimized fluence intensity map (IM) for each beam is generated via inverse planning, this IM is then segmented into discrete levels, and finally, the segmented map is translated into a set of MLC apertures via a leaf sequencing algorithm. To date, limited work has been done on this approach as it pertains to intensity modulated arc therapy (IMAT), specifically in regards to the latter two steps. There are two determining factors that separate IMAT segmentation and leaf sequencing from their IMRT equivalents: (1) the intrinsic 3D nature of the intensity maps (standard 2D maps plus the angular component), and (2) that the dynamic multileaf collimator (MLC) constraints be met using a minimum number of arcs. In this work, we illustrate a technique to create an IMAT plan that replicates Tomotherapy deliveries by applying IMAT specific segmentation and leaf-sequencing algorithms to Tomotherapy output sinograms. We propose and compare two alternative segmentation techniques, a clustering method, and a bottom-up segmentation method (BUS). We also introduce a novel IMAT leaf-sequencing algorithm that explicitly takes leaf movement constraints into consideration. These algorithms were tested with 51 angular projections of the output leaf-open sinograms generated on the Hi-ART II treatment planning system (Tomotherapy Inc.). We present two geometric phantoms and 2 clinical scenarios as sample test cases. In each case 12 IMAT plans were created, ranging from 2 to 7 intensity levels. Half were generated using the BUS segmentation and half with the clustering method. We report on the number of arcs produced as well as differences between Tomotherapy output sinograms and segmented IMAT intensity maps. For each case one plan for each segmentation method is chosen for full Monte Carlo dose calculation (NumeriX LLC) and dose volume histograms (DVH) are calculated. In all cases, the BUS method outperformed the clustering, method. We recommend using the BUS algorithm and discuss potential improvements to the clustering algorithms.

Links between the Slater Index and the Ryser Index of Tournaments

Given a tournament T, the Slater index i(T) of T is the minimum number of arcs that must be reversed to make T transitive. The Ryser index τ˜(T) of T, defined from the out-degrees of T, measures a remoteness between T and the transitive tournaments of same order. In this paper, we study some links between i(T) and τ˜(T). More precisely, calling I(n, τ) the maximum value of i(T) over the set of tournaments on n vertices and such that τ˜(T)=τ, we compute an upper bound of I(n,τ) for every value of τ. Then we use this upper bound to study a conjecture stated by J.-C. Bermond on the regularity (i.e., the fact that all the out-degrees are equal or almost equal) of the tournaments with a maximum Slater index by showing that the out-degrees of such tournaments cannot be “too far” from the ones of the regular tournaments.

Strongly Connected Spanning Subdigraphs with the Minimum Number of Arcs in Quasi-transitive Digraphs

We consider the problem of finding a strongly connected spanning subdigraph with the minimum number of arcs in a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the Hamiltonian cycle problem. We show that the problem is polynomially solvable for quasi-transitive digraphs. We describe the minimum number of arcs in such a spanning subdigraph of a quasi-transitive digraph in terms of the path covering number. Our proofs are based on a number of results (some of which are new and interesting in their own right) on the structure of cycles and paths in quasi-transitive digraphs and in extended semicomplete digraphs. In particular, we give a new characterization of the longest cycle in an extended semicomplete digraph. Finally, we point out that our proofs imply that the MSSS problem is solvable in polynomial time for all digraphs that can be obtained from strong semicomplete digraphs on at least two vertices by replacing each vertex with a digraph belonging to a family of digraphs whose path covering number can be decided in polynomial time.