Bipartite graphs Research Papers - Academia.edu (original) (raw)

2025, MATCH Communications in Mathematical and in Computer Chemistry

We show that if the radius of a simple, connected graph equals its indepen-dence number, then the graph contains a Hamiltonian path. This result was conjectured by the computer program Graffiti.pc, using a new conjecture-generating strategy called Sophie. We also mention several other sufficient conditions for Hamiltonian paths that were conjectured by Graffiti.pc, but which are currently open, so far as we know.

2025, Discrete Mathematics

For integers p; q; s with p¿q¿3 and 16s6q -1, let K -s (p; q) (resp. K -s 2 (p; q)) denote the set of connected (resp. 2-connected) bipartite graphs which can be obtained from Kp;q by deleting a set of s edges. In this paper, we ÿrst ÿnd an upper bound for the 3-independent partition number of a graph G ∈ K -s (p; q) with respect to the maximum degree (G ) of G , where G = Kp;q -G. By using this result, we show that the set {G | G ∈ K -s 2 (p; q); (G ) = i} is closed under the chromatic equivalence for every integer i with s¿i¿(s + 3)=2. From this result, we prove that for any G ∈ K -s 2 (p; q) with p¿q¿3, if 56s6q -1 and (G ) = s -1, or 76s6q -1 and (G ) = s -2, then G is chromatically unique.

2025, Lecture Notes in Computer Science

We study an online model for the maximum k-vertex-coverage problem, where given a graph G = (V, E) and an integer k, we ask for a subset A ⊆ V , such that |A| = k and the number of edges covered by A is maximized. In our model, at each step i, a new vertex vi is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k vertices can be kept in memory; if at some point the current solution already contains k vertices, any inclusion of any new vertex in the solution must entail the irremediable deletion of one vertex of the current solution (a vertex not kept when revealed is irremediably deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy 1 2 -competitive ratio. We next settle a set-version of the problem, called maximum k-(set)-coverage problem. For this problem we present an algorithm that improves upon former results for the same model for small and moderate values of k.

2025, Discrete Mathematics

Let F(n, e) be the collection of all simple graphs with n vertices and e edges, and for G ∈ F(n, e) let P (G; ) be the chromatic polynomial of G. A graph G ∈ F(n, e) is said to be optimal if another graph H ∈ F(n, e) does not exist with P (H ; ) P (G; ) for all , with strict inequality holding for some . In this paper we derive necessary conditions for bipartite graphs to be optimal, and show that, contrarily to the case of lower bounds, one can find values of n and e for which optimal graphs are not unique. We also derive necessary conditions for bipartite graphs to have the greatest number of cycles of length 4.

2025, Discrete Mathematics

2025, Explore various optimization problems including Traveling Salesperson Problem (TSP), the Knapsack Problem, and the Assignment Problem

This research paper explores various optimization problems, including the Traveling Salesperson Problem (TSP), the Knapsack Problem, and the Assignment Problem. Each problem is analyzed in terms of its concepts, applications, solution approaches, and complexities. The paper aims to provide a comprehensive understanding of these but not for publishing publicly, just for the internal usage only.

2024

We extend the definition of sandwich line-graphs, a class of auxiliary graphs the stable sets of which are in 1-to-1 correspondence with the colorings of the original graph, from graphs to partitioned graphs, this way, we obtain a... more

2024, Southeast Asian Bulletin of Mathematics,

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, • • • , 2q − 1} such that the induced function f * : E(G) → {1, 3, • • • , 2q − 1} defined by f * (uv) = f (u) + f (v) is a bijection. In this paper we prove that m-shadow and m-splitting of the graphs Pn, Hn,n, Kr,s, Pn ⊕ K2 and Splm(Cn), n ≡ 0(mod 4) are odd harmonious graphs.

2024, Australas. J Comb.

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008), 219– 222] presented a constructive characterization of bipartite graphs G of order n for which α(G) = n 2 . In this paper we characterize all bipartite graphs G of order n with α(G) = k, for each n 2 ≤ k ≤ n − 1. We also give a characterization on the Nordhaus-Gaddum type inequalities on the transversal number of trees.

2024, Discrete Mathematics

Borowiecki, M. and E. Drgas-Burchardt, Classes of chromatically unique graphs, Discrete Mathematics Ill (1993) 71-75. We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are... more

2024, arXiv (Cornell University)

We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.

2024, Australas. J Comb.

For a graph G = (V, E), a non-empty set S ⊆ V is a global offensive alliance if for every v ∈ V − S, at least half of the vertices from the closed neighborhood of v are in S. A set S ⊆ V is a global strong offensive alliance if for each vertex v ∈ V − S, a strict majority of the vertices of the closed neighborhood of v are in S. The cardinality of a minimum global (strong) offensive alliance of a graph G is called the global (strong) offensive alliance number of G. We determine bounds on the global offensive alliance and the global strong offensive alliance numbers of a graph, and characterize the trees achieving two of these lower bounds.

2024, HAL (Le Centre pour la Communication Scientifique Directe)

2024, Discrete Applied Mathematics

2024

This paper presents an algorithm, based on the fixed point iteration, to solve for sizes and biases using transistor compact models such as BSIM3v3, BSIM4, PSP and EKV. The proposed algorithm simplifies the implementation of sizing and biasing operators. Sizing and biasing operators were originally proposed in the hierarchical sizing and biasing methodology . They allow to compute transistors sizes and biases based on transistor compact models, while respecting the designer's hypotheses. Computed sizes and biases are accurate, and guarantee the correct electrical behavior as expected by the designer. Sizing and biasing operators interface with a Spice-like simulator, allowing possible use of all available compact models for circuit sizing and biasing over different technologies. A bipartite graph, that contains sizing and biasing operators, is associated to the design view of a circuit, it is the design procedure for the given circuit. To illustrate the effectiveness of the proposed fixed point algorithm, a folded cascode OTA is efficiently sized with a 130nm process, then migrated to a 65nm technology. Both sizing and migration are performed in a few milliseconds.

2024, Theoretical Computer Science

A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a "ground line," a line with slope −1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G's bipartite adjacency matrix with rows A and columns B excludes three small 'forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A| 3 |B| 3)-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.

2024, Computing and Informatics / Computers and Artificial Intelligence

Automatic synthesis of control for a kind of DES (discrete-event systems) is discussed and an approach to it is proposed and presented. The approach consists in the proposal of the control synthesis procedure based on bipartite directed graphs yielding both the feasible control trajectories and the corresponding state ones. Soundness of the approach is tested on examples. Then, the usage of the approach is combined with the supervisor synthesis in order to complement it. Applicability of such approach is demonstrated by means of several illustrative examples of both the single agents and the agent cooperation in MAS.

2024, RePEc: Research Papers in Economics

In the standard independent private values (IPV) model, each bidder's beliefs about the values of any other bidder is represented by a unique prior. In this paper we relax this assumption and study the question of auction design in an IPV setting characterized by ambiguity: bidders have an imprecise knowledge of the distribution of values of others, and are faced with a set of priors. We also assume that their preferences exhibit ambiguity aversion. We show that a simple variation of a discrete Dutch auction can extract almost all surplus. This contrasts with optimal auctions under IPV without ambiguity as well as with optimal static auctions with ambiguity-in all of these, types other than the lowest participating type obtain a positive surplus. And, unlike the well-known Cremer-McLean mechanism, our modified Dutch mechanism satisfies limited liability. An important point of departure is that the modified Dutch mechanism we consider is dynamic rather than static, establishing that under ambiguity aversion-even when the setting is IPV in all other respects-a dynamic mechanism could have additional bite over its static counterparts.

2024, Journal of Combinatorial Theory, Series B

F. Bry (J. Combin. Theory Ser. B 34 (1983). 48-57) proved that a locally finite infinite n-connected factorizable graph has at least (n-l)! l-factors and showed that for n =2 this lower bound is sharp. We prove that for n 33 any infinite... more

2024, Journal of Combinatorial Theory, Series B

2024, HAL (Le Centre pour la Communication Scientifique Directe)

2024, Journal of Graph Theory

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let 𝒦−s(p,q) (resp., 𝒦2−s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ 𝒦 −s(p,q) and 0 ≤ s ≤ (p − 1)(q − 1) (resp., G ∈ 𝒦 2−s(p,q) and 0 ≤ s ≤ p + q − 4). These bounds are then used to show that if G ∈ 𝒦 −s(p,q) (resp., G ∈ 𝒦 2−s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets 𝒦−si(p+i,q−i) where max −soverq−1,−p−qover2!le,ilesoverp−1\{-{s\over q-1},-{p-q\over 2}\}\!\le \,i \le{s\over p-1}−soverq−1,−p−qover2!le,ilesoverp−1 and si = s − i(p−q+i) (resp., a subset of 𝒦2−s(p,q), where either 0 ≤ s ≤ q − 1, or s ≤ 2q − 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q − 1 or that induces a star is chromati...

2024, Chemical engineering transactions

The P-graph framework was originally developed to address Process Network Synthesis (PNS) problems in the preliminary design of chemical plants. P-graph provides a mathematically rigorous and computationally efficient framework for solving PNS problems via the maximal structure generation (MSG), solution structure generation (SSG) and accelerated branch-and-bound (ABB) algorithms. MSG ensures rigorous generation of the maximal structure, while the ad hoc generation of a superstructure as basis for a mathematical programming model can lead to significant modelling errors. In addition, SSG allows the generation of combinatorially feasible network structures that can be utilized for practical decision-making by designers. For very large problems, ABB can reduce the computational effort of reaching globally optimal solutions by multiple orders of magnitude compared to conventional branch-and-bound solvers for Mixed Integer Linear Programming (MILP) models. In addition to conventional PN...

2024, Rairo-operations Research

A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecting hyperedges, has a common vertex. We consider the concepts of bipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way as conformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal and bipartite-Helly hypergraphs are also dual. They are useful for characterizing biclique matrices and biclique graphs, that is, the incident biclique-vertex incidence matrix and the intersection graphs of the maximal bicliques of a graph, respectively. These concepts play a similar role for the bicliques of a graph, as do clique matrices and clique graphs, for the cliques of the graph. We describe polynomial time algorithms for recognizing bipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.

2024

The adjacency matrix A of a graph G is a 0-1 matrix. The Boolean power sequence of A is convergent or periodic of period p = 2. The index γ of A is the least integer m such that A m = A m+1 if A converges and the least integer m such that A m = A m+2 if A is periodic. In this paper we determine the index γ of A if the graph G is bipartite. In the case of non-bipartite connected graphs, we give new lower and upper bounds for γ , which are sharp.

2024, arXiv (Cornell University)

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph G, in terms of the cardinality of a vertex cover of minimum size in G containing all its cut vertices. The consequences of the lower bound includes a... more

2024, 2015 IEEE Computer Society Annual Symposium on VLSI

Complex graphs are at the heart of today's big data challenges like recommendation systems, customer behavior modeling, or incident detection systems. One reoccurring task in these fields is the extraction of network motifs, reoccurring and statistically significant subgraphs. In this work we propose a precisely tailored embedded architecture for computing similarities based on one special network motif, the co-occurrence. It is based on efficient and scalable building blocks that exploit well-tuned algorithmic refinements and an optimized graph data representation approach. On chip, our solution features a customized cache design and a lightweight data path that allows the system to perform over 10,000 graph operations per cycle on each chip. We provide detailed area, energy, and timing results for a 28 nm ASIC process and DDR3 memory devices. Compared to an Intel cluster, our proposed solution uses 44x less memory and is 224x more energy efficient.

2024, Discrete Mathematics

For integers p; q; s with p¿q¿3 and 16s6q − 1, let K −s (p; q) (resp. K −s 2 (p; q)) denote the set of connected (resp. 2-connected) bipartite graphs which can be obtained from Kp;q by deleting a set of s edges. In this paper, we ÿrst ÿnd an upper bound for the 3-independent partition number of a graph G ∈ K −s (p; q) with respect to the maximum degree (G) of G , where G = Kp;q − G. By using this result, we show that the set {G | G ∈ K −s 2 (p; q); (G) = i} is closed under the chromatic equivalence for every integer i with s¿i¿(s + 3)=2. From this result, we prove that for any G ∈ K −s 2 (p; q) with p¿q¿3, if 56s6q − 1 and (G) = s − 1, or 76s6q − 1 and (G) = s − 2, then G is chromatically unique.

2024, arXiv (Cornell University)

Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graphtheoretic conditions on the set of circuits of G.

2024, arXiv (Cornell University)

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.

2024, Časopis pro pěstování matematiky

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2024

When working on graphs, reachability is among the most common problems to address, since it is the base for many other algorithms. As with the advent of distributed systems, which process large amounts of data, many applications must quickly explore graphs with millions of vertices, scalable solutions have become of paramount importance. Modern GPUs provide highly parallel systems based on many-core architectures and have gained popularity in parallelizing algorithms that run on large data sets. In this paper, we extend a very efficient state-of-the-art graph-labeling method, namely the GRAIL algorithm, to architectures which exhibit a great amount of data parallelism, i.e., many-core CUDA-based GPUs. GRAIL creates a scalable index for answering reachability queries, and it heavily relies on depth-first searches. As depth-first visits are intrinsically recursive and they cannot be efficiently implemented on parallel systems, we devise an alternative approach based on a sequence of breadth-first visits. The paper explores our efforts in this direction, and it analyzes the difficulties encountered and the solutions chosen to overcome them. It also presents a comparison (in terms of times to create the index and to use it for reachability queries) between the CPU and the GPUbased versions.

2023, Springer eBooks

We consider extension variants of Vertex Cover and Independent Set, following a line of research initiated in [9]. In particular, we study the Ext-CVC and the Ext-NSIS problems: given a graph G = (V, E) and a vertex set U ⊆ V , does there exist a minimal connected vertex cover (respectively, a maximal non-separating independent set) S, such that U ⊆ S (respectively, U ⊇ S). We present hardness results for both problems, for certain graph classes such as bipartite, chordal and weakly chordal. To this end we exploit the relation of Ext-CVC to Ext-VC, that is, to the extension variant of Vertex Cover. We also study the Price of Extension (PoE), a measure that reflects the distance of a vertex set U to its maximum efficiently computable subset that is extensible to a minimal connected vertex cover, and provide negative and positive results for PoE in general and special graphs.

2023, Graphs and Combinatorics

Finding the multiplicity of cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. Recently, Blake and Lin computed the number of shortest cycles (g-cycles, where g is the girth of the graph) in a bi-regular bipartite graph, in terms of the degree sequences and the spectrum (eigenvalues of the adjacency matrix) of the graph [IEEE Trans. Inform. Theory 64(10):6526-6535, 2018]. This result was subsequently extended in [IEEE Trans. Inform. Theory, accepted for publication, Dec. 2018] to cycles of length g + 2,. .. , 2g − 2, in bi-regular bipartite graphs, as well as 4-cycles and 6-cycles in irregular and halfregular bipartite graphs, with g ≥ 4 and g ≥ 6, respectively. In this paper, we complement these positive results with negative results demonstrating that the information of the degree sequences and the spectrum of a bipartite graph is, in general, insufficient to count (a) the i-cycles, i ≥ 2g, in bi-regular graphs, (b) the i-cycles for any i > g, regardless of the value of g, and g-cycles for g ≥ 6, in irregular graphs, and (c) the i-cycles for any i > g, regardless of the value of g, and g-cycles for g ≥ 8, in half-regular graphs. To obtain these results, we construct counterexamples using the Godsil-McKay switching.

2023, Journal of Combinatorial Theory, Series B

We consider monotone embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l n 2 , while, (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3). It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space-that defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n/(λ 2 + 1)), where λ 2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0 < δ ≤ 1 2 is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quasi-random graphs of logarithmic sphericity (Lemma 7). For the above bound to be linear, λ 2 must be constant. We show that if the second eigenvalue of an n/2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n 2) entries (Theorem 5). Furthermore, for any 0 < δ < 1 2 , and λ 2 , there are only finitely many δn-regular graphs with second eigenvalue at most λ 2 (Corollary 4).

2023, International journal of mathematics and soft computing

The all-ones problem is an NP-complete problem introduced by Sutner [11], with wide applications in linear cellular automata. In this paper, we solve the all-ones problem for some of the widely studied architectures like binomial trees,... more

2023, Journal of Economic Theory

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2023, Discrete Mathematics

Given a simple and finite connected graph G, the distance d G (u, v) is the length of the shortest induced {u, v}-path linking the vertices u and v in G. Bandelt and Mulder [H.J. Bandelt, H.M. Mulder, Distance hereditary graphs, J. Combin. Theory Ser. B 41 (1986) 182-208] have characterized the class of distance hereditary graphs where the distance is preserved in each connected induced subgraph. In this paper, we are interested in the class of k-distance hereditary graphs (k ∈ N) which consists in a parametric extension of the distance heredity notion. We allow the distance in each connected induced subgraph to increase by at most k. We provide a characterization of k-distance hereditary graphs in terms of forbidden configurations for each k ≥ 2.

2023, Discrete Mathematics

The notion of distance-heredity in graphs has been extended to construct the class of almost distance-hereditary graphs (an increase of the distance by one unit is allowed by induced subgraphs). These graphs have been characterized in terms of forbidden induced subgraphs [M. Aïder, Almost distance-hereditary graphs, Discrete Math. 242 (1-3) (2002) 1-16]. Since the distance in bipartite graphs cannot increase exactly by one unit, we have to adapt this notion to the bipartite case. In this paper, we define the class of bipartite almost distance-hereditary graphs (an increase of the distance by two is allowed by induced subgraphs) and obtain a characterization in terms of forbidden induced subgraphs.

2023, Discrete Mathematics

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.

2023, Discrete Mathematics

In a graph G = (V , E) of order n and maximum degree , a subset S of vertices is a k-independent set if the subgraph induced by S has maximum degree less or equal to k − 1. The lower k-independence number i k (G) is the minimum cardinality of a maximal k-independent set in G and the k-independence number k (G) is the maximum cardinality of a k-independent set. We show that i k n − + k − 1 for any graph and any k , and i 2 n − if G is connected, that k (T) kn/(k + 1) for any tree, and that i 2 (n + s)/2 2 for any connected bipartite graph with s support vertices. Moreover, we characterize the trees satisfying i 2 = n − , k = kn/(k + 1), i 2 = (n + s)/2 or 2 = (n + s)/2.

2023, Discussiones Mathematicae Graph Theory

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted i k (G) and β k (G). We give some relations between β k (G) and β j (G) and between i k (G) and i j (G) for j = k. We study two families of extremal graphs for the inequality i 2 (G) ≤ i(G) + β(G). Finally we give an upper bound on i 2 (G) and a lower bound when G is a cactus.

2023, Discrete Mathematics

We determine upper bounds on the ratios of several domination parameters in trees.

2023, Discrete Mathematics

We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and oriented edges. This problem is related to a communication problem in job-shop scheduling systems. In this paper we give general bounds on the number of required colors and analyze the complexity status of this problem. In particular, we provide N Pcompleteness results for the case of outerplanar graphs, as well as for 3-regular bipartite graphs (even when only 3 colors are allowed, or when 5 colors are allowed and the graph is fully oriented). Special cases admitting polynomial-time solutions are also discussed.

2023, Discrete Applied Mathematics

In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s 1 , s 2 , and s 3 to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in O(n 2) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s 1 > s 2 = s 3. However, in this case there exists a 10/7-approximation algorithm running in O(n 3) time. Moreover, this algorithm solves the problem almost surely to optimality if 3s 1 /4 ≤ s 2 = s 3 .

2023, Networks

Consider a network G = (V,E) with distinguished vertices s and t, and with k different costs on every edge. We consider the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized, where the jth edge‐cost is associated with the jth path. The problem has several variants: The paths may be vertex‐disjoint or arc‐disjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP‐complete even for k = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.

2023, arXiv (Cornell University)

Let G = (V, E) be a simple graph. A non-empty set S ⊆ V is called a global offensive alliance if S is a dominating set and for every vertex v in V − S, at least half of the vertices from the closed neighborhood of v are in S. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.

2023, arXiv (Cornell University)

For a graph G = (V, E), a set S ⊆ V is a dominating set if every vertex in V − S has at least a neighbor in S. A dominating set S is a global offensive alliance if for each vertex v in V − S at least half the vertices from the closed neighborhood of v are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We show that if G is a connected unicycle graph of order n with l(G) leaves and s(G) support vertices then γo(G) ≥ n−l(G)+s(G) 3. Moreover, we characterize all extremal unicycle graphs attaining this bound.

2023, Theoretical Computer Science

2023, Discrete Mathematics & Theoretical Computer Science

We draw the r-dimensional butterfly network with 1 4 4 r +O(r2 r) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper.