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Lossless text compression by means of binary-coded ternary number representation Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-14 Igor O. Zavadskyi
A lossless data compression code based on a binary-coded ternary number representation is investigated. The code is complete, universal, synchronizable, and supports a direct search in a compressed file. A simple monotonous encoding and very fast decoding algorithms are constructed owing to code properties. Experiments show that in natural language text compression, the new code outperforms byte-aligned
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Removable and forced subgraphs of graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-13 Wuxian Chen, Heping Zhang
A connected graph is - if is a subgraph of , has at least vertices and has a perfect matching for every subgraph of isomorphic to . So a connected graph is -removable if and only if it is -extendable, where is a matching of size . Further is an - if has a perfect matching for every subgraph of isomorphic to . In this paper, we first characterize -forced graphs for as and by using randomly matchable
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A value for cooperative games on simplicial complexes with a filtration Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-12 J.C. Rodríguez-Gómez, Manuel Ordóñez Sánchez, A. Jiménez-Losada
The classical Shapley value for cooperative games determines a payoff vector considering that the formation of the grand coalition is made by incorporating players one by one. Later, this method was generalized for games with restricted cooperation by several known mathematical structures: partitions, graphs, convex geometries, antimatroids, matroids or simplicial complexes. In this paper we consider
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Metric spaces in which many triangles are degenerate Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-11 Vašek Chvátal, Noé de Rancourt, Guillermo Gamboa Quintero, Ida Kantor, Péter G.N. Szabó
Richmond and Richmond (1997) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In a metric space on points, fewer than suitably placed degenerate triangles suffice. However, fewer than degenerate triangles, no matter how cleverly
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Edge-disjoint properly colored cycles in edge-colored complete graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-09 Xiaozheng Chen, Luyi Li, Xueliang Li
In an edge-colored graph , let denote the maximum number of edges with the same color incident with a vertex in , called the of . The maximum value of over all vertices is called the of , denoted by . Li et al. in 2019 conjectured that every edge-colored complete graph of order with contains vertex-disjoint properly colored (PC for short) cycles of length at most 4, and they showed that the conjecture
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Pure strategy solutions of the progressive discrete silent duel with generalized identical quadratic accuracy functions Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-08 Vadim V. Romanuke
A generalized class of the discrete game of timing is solved as a finite zero-sum game defined on a symmetric lattice of the unit square. The game is a progressive discrete silent duel whose kernel is skew-symmetric, and the players, referred to as duelists, have identical shooting accuracy functions featured with an accuracy proportionality factor and a power constant describing nonlinearity of the
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An impossible combinatorial counting method in distance geometry Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-05 Germano Abud, Jorge Alencar, Carlile Lavor, Leo Liberti, Antonio Mucherino
The Distance Geometry Problem asks for a geometric representation of a given weighted graph in so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on -lateration: find two possible positions for the next vertex using the positions
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On the diameter of Schrijver graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-05 Agustina Victoria Ledezma, Adrián Pastine, Pablo Torres, Mario Valencia-Pabon
For and , the well known Kneser graph has all -element subsets of an -element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph of with the same chromatic number. In this paper, we compute the diameter of the graph with . We obtain an exact value of the diameter of when or when . For the remaining cases, when , we show that the diameter
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Integral values of generating functions of recursive sequences Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-04 M. Knapp, A. Lemos, Victor G.L. Neumann
Suppose that is an integer sequence which satisfies a recurrence relation with constant coefficients, and let be its generating function, where and have no common factors in . In this article, we study the problem of finding the rational values of such that is an integer. We say that such a number is for the sequence. Our first main result is that if has at least two different irreducible factors,
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Proving a conjecture on prime double square tiles Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-04 Michela Ascolese, Andrea Frosini
In 2013, while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, Blondin Massé et al. found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have specific interesting properties that involve notions of combinatorics on words such as palindromicity, periodicity and symmetry. Furthermore, they defined
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The axiomatic characterization of the interval function of distance hereditary graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-04 Manoj Changat, Lekshmi Kamal Kamalolbhavan-Sheela, Prasanth G. Narasimha-Shenoi
A connected graph is distance hereditary if every induced path in is a shortest path. The of a connected graph is defined as the set of all vertices that lie on some shortest -path in . In this paper, we consider certain types of first-order betweenness axioms framed on an arbitrary function known as and used to characterize the interval function of a distance hereditary graph. As a byproduct, we give
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A worst-case optimal algorithm to compute the Minkowski sum of convex polytopes Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-04 Sandip Das, Subhadeep Ranjan Dev, Sarvottamananda
We propose algorithms to compute the Minkowski sum of two or more convex polytopes represented by their face lattices in . The time and space complexities of the pair-wise algorithm are and , respectively, where , and are the face lattice sizes of the two summands and the sum, respectively, and , currently, is the matrix multiplication exponent. We also show that this algorithm is worst-case optimal
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Cycle intersection in spanning trees: A shorter proof of a conjecture and applications Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-04 P, a, b, l, o, , D, e, , C, a, r, i, a, , D, i, , F, o, n, z, o
Consider a connected simple graph . Given a spanning tree of , for each edge in but not in , a cycle is formed by adding the edge to the path in that connects the endpoints of . The Minimum Spanning Tree Cycle Intersection problem (MSTCI for short) consists in finding a spanning tree for that minimizes the number of intersections between this type of cycles. This problem was introduced in 2021 and
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Branchwidth is [formula omitted]-self-dual Discrete Appl. Math. (IF 1.1) Pub Date : 2024-03-01 Georgios Kontogeorgiou, Alexandros Leivaditis, Kostas I. Psaromiligkos, Giannos Stamoulis, Dimitris Zoros
A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that, in the class of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most , branchwidth is a -self-dual parameter, i.e., for every hypergraph in the class, the branchwidth of its
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Recognizing single-peaked preferences on an arbitrary graph: Complexity and algorithms Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-23 Bruno Escoffier, Olivier Spanjaard, Magdaléna Tydrichová
We study in this paper single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of alternatives), we aim at determining a connected graph on which the preferences are single-peaked, in the sense that all the preferences are traversals of . Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation
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Extremal Kirchhoff index in polycyclic chains Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-15 Hechao Liu, Lihua You
The Kirchhoff index of a graph is defined as , where denotes the resistance distance between and in . In this paper, we determine the maximum (resp. minimum) -polycyclic chains with respect to Kirchhoff index for , which extends the results of Yang and Klein (2014), Yang and Sun (2022), Sun and Yang (2023) and Ma (2022).
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Roman domination and independent Roman domination on graphs with maximum degree three Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-15 Atílio G. Luiz
A Roman dominating function (RDF) on a graph is a function such that every vertex with is adjacent to at least one vertex with . An independent Roman dominating function (IRDF) on a graph is a Roman dominating function such that the set of vertices with form an independent set on . The weight of is . The minimum weight of an RDF (IRDF) on is the Roman domination number (independent Roman domination
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Some sufficient conditions for a graph with minimum degree to be [formula omitted]-factor-critical Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-15 Lin Zheng, Shuchao Li, Xiaobing Luo, Guangfu Wang
A graph is said to be -factor-critical if deleting any of its vertices results in a graph with a perfect matching. In this paper, we establish three sufficient conditions for a graph with given minimum degree to be -factor-critical. These three sufficient conditions include the following: the size condition, -index condition and distance spectral radius condition.
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Values for cooperative games with a prior unions and a communication graph based on combined effects Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-14 Rong Zou, Genjiu Xu, Wenzhong Li, Panfei Sun
In this paper we focus on restricted cooperation modeled by TU-games with a prior unions and a communication graph. We deal with the two structures in a comprehensive way by considering two potential combined effects of them, called integration and limitation. While the former makes it possible for sets of players who are unconnected in the graph but intersect with same unions to cooperate, the latter
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The median function of a block graph: Axiomatic characterizations Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-12 Manoj Changat, Gokul Krishna Gopakumar-Sheejakumari, Prasanth G. Narasimha-Shenoi
A profile of length in a connected graph is a sequence of vertices in with possible repetitions of vertices. A median of a profile in is a vertex that minimizes the remoteness value, that is, the sum of the distances from to the elements in is minimized. The median function output the set of medians (denoted as ) of , for every profile in . It is one of the basic models for the location of a desirable
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On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-06 Aida Abiad, Robin Simoens, Sjanne Zeijlemaker
We determine the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results.
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Algebraic properties of soluble posets Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-06 Grega Cigler, Piotr J. Wojciechowski
In this note we exhibit various ways of obtaining a wide range of soluble partially ordered sets. These are the posets on which every transitive system can be completed. In particular, algebraic operations such as the sums and products of posets are proven to be powerful tools to create new soluble posets.
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On the sizes of generalized cactus graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-05 Licheng Zhang, Yuanqiu Huang
A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a -cactus is a connected graph in which each edge is contained in at most cycles where . It is well known that every cactus with vertices has at most edges. Inspired by it, we attempt to establish analogous upper bounds for general -cactus graphs. In this paper, we first
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One-sided terrain guarding and chordal graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-05 Prahlad Narasimhan Kasthurirangan
The problem, a variant of the famous problem, has garnered significant attention over the last two decades in Computational Geometry from the viewpoint of complexity and approximability. Both the continuous and discrete versions of the problem were shown to be NP-Hard in King and Krohn (2010) and to admit a PTAS (Krohn et al., 2014; Friedrichs et al., 2016). The biggest unsolved question regarding
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On CD-chromatic number and its lower bound in some classes of graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-05 Shalu M.A., Kirubakaran V.K.
A -class domination coloring (-cd-coloring) is a partition of the vertex set of a graph into independent sets , where each is dominated by some vertex of . The least integer such that admits a -cd-coloring is called the cd-chromatic number, , of . A subset of the vertex set of a graph is called a subclique in if for every . The cardinality of a maximum subclique in is called the subclique number,
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A note on the conditional fault-tolerant strong Menger edge connectivity of regular graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-03 Pingshan Li, Min Xu, Eddie Cheng
A graph is said to be strongly Menger edge-connected (SM- for short) if for every two of its vertices and , there are ) edge-disjoint paths between them, where is the degree of in . The maximum conditional edge-fault-tolerant number of order with respect to the SM- property of , denoted by , is the maximum integer such that remains SM- for every edge set with and . So far, most of the exact values
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Divisibility and coloring of some [formula omitted]-free graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-03 Jialei Song, Baogang Xu
A is a path on 5 vertices, a banner is a graph obtained by adding a pendant edge to a vertex of a quadrilateral and a hammer is a graph obtained from a by deleting a banner as a partial subgraph. A graph is perfect if for each induced subgraph of . We say that admits a perfect division if can be partitioned into two subsets and such that is perfect and , and say that admits a 2-division if or can be
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A new note on 1-planar graphs with minimum degree 7 Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-03 Yuanqiu Huang, Licheng Zhang, Fengming Dong
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is well-known that each 1-planar graph has its minimum degree at most 7. Recently, Biedl (2021) showed that any 1-planar graph with minimum degree 7 has at least 24 vertices. In this paper, we characterize 1-planar graphs with 24 vertices and minimum degree 7. Furthermore, we prove that any 1-planar
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The list [formula omitted]-hued coloring of [formula omitted] Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-03 Meng Tang, Fengxia Liu, Hong-Jian Lai
Let be list assignment of colors available for vertices of a graph . An -coloring of is a proper coloring such that for any vertex , we have and min. The list -hued chromatic number of , denoted as , is the least integer , such that for list assignment satisfying , for any , has an -coloring. Let denote a complete bipartite graph. In [Discrete Math. 306(16)(2006) 1997–2004], it has been proved if
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The minimum ABC index of chemical trees Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-02 Wei Gao
Let be a graph with vertex set and edge set . The ABC index of , denoted by , is defined as In [B. Furtula, A. Graovac, D. Vukičević, Atom-bond connectivity index of trees, Discrete Appl. Math. 157 (2009) 2828–2835], the authors gave a lower bound of the ABC index for chemical trees. Noticing that this lower bound is not consistently tight, we completely determine the minimum ABC index for chemical
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Edge deletion to tree-like graph classes Discrete Appl. Math. (IF 1.1) Pub Date : 2024-02-02 Ivo Koch, Nina Pardal, Vinicius Fernandes dos Santos
For a fixed property (graph class) , given a graph and an integer , the - problem consists in deciding if we can turn into a graph with the property by deleting at most edges. The -deletion problem is known to be NP-hard for most of the well-studied graph classes, such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others; even deletion to is known to be NP-hard
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The [formula omitted]-analogue of zero forcing for certain families of graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-28 Shaun Fallat, Neha Joshi, Roghayeh Maleki, Karen Meagher, Seyed Ahmad Mojallal, Shahla Nasserasr, Mahsa N. Shirazi, Andriaherimanana Sarobidy Razafimahatratra, Brett Stevens
Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a one player game, but later a two-player version was devised in-conjunction with studies on the inertia of a graph, and has become known as the -analogue of zero forcing. In this paper, we study and compute the -analogue zero
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Listing the bonds of a graph in [formula omitted]–delay Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-26 Alice Raffaele, Romeo Rizzi, Takeaki Uno
Given a connected graph , with vertices and edges, a cut can be represented as a bipartition of the vertices or as the set of those edges in having one endpoint in and the other in , denoted by . A cut is minimal, or also called , if and only if the two induced subgraphs obtained by removing the edges in the cut are both connected. When the bond separates two given vertices and , we talk about -. In
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Graph curvature via resistance distance Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-26 Karel Devriendt, Andrea Ottolini, Stefan Steinerberger
Let be a finite, combinatorial graph. We define a notion of curvature on the vertex set via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by have diameter bounded from above. The Laplacian satisfies a Lichnerowicz estimate, there is a spectral gap . We obtain matching two-sided bounds
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On the eccentric distance sum of trees with given maximum degree Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-26 Ting Zhou, Lianying Miao, Wenyao Song
Let be a simple connected graph. The eccentric distance sum (EDS) of is defined as where is the eccentricity of the vertex and is the sum of all distances from the vertex We denote the set of trees with order and maximum degree by . In 2015, the tree having the maximal EDS among all trees in was determined (Miao, 2015). In this paper, the tree having the second maximal EDS among all trees in is characterized
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New characterizations and a concept of potential for each multinomial (probabilistic) value Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-25 Margarita Domènech, José Miguel Giménez, María Albina Puente
In this paper we focus on multinomial probabilistic values and we consider two special classes of players: necessary and nullifying players. By introducing new properties related to this kind of players, we provide new axiomatic characterizations of each multinomial probabilistic value, giving, in all cases, a set of independent properties that univocally determine them. Moreover, when the profile
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Concerning a conjecture on matching Kneser graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-25 Saeed Shaebani
Alishahi and Hajiabolhassan found that for some classes of graphs, the wonderful equality holds as an amazing relationship between chromatic number and generalized Turán number. This powerful equality enabled them to determine chromatic numbers of some interesting and important classes of graphs. They conjectured that the aforementioned equality holds for all connected graphs . Iradmusa, by a nice
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Some graphs determined by their [formula omitted]-spectra Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-23 Yuanyuan Chen, Dan Li, Huiqiu Lin, Jixiang Meng
For any real , define the matrix as The collection of eigenvalues of together with multiplicities is called the -spectrum of . A graph is said to be determined by its -spectrum if any graph that is -cospectral with is isomorphic to . Lin et al. (2019) proposed characterizing graphs determined by their -spectra such that are also determined by their -spectra for . In this paper, we get that if , then
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Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-19 Stijn Cambie
In the original article (Gutman et al., 2014), the authors state that the Wiener index (total distance) of an Eulerian graph is maximized by the cycle. We explain that the initial proof contains a flaw and note that it is a corollary of a result by Plesník, since an Eulerian graph is 2-edge-connected. The same incorrect proof is used in two referencing papers, (Liu et al., 2019) and (Cai et al., 2021)
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The Maker–Maker domination game in forests Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-22 Eric Duchêne, Arthur Dumas, Nacim Oijid, Aline Parreau, Eric Rémila
We study the Maker–Maker version of the domination game introduced in 2018 by Duchêne . Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker–Breaker version, this game is -complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the
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Maxima of the [formula omitted]-index of graphs with given size and domination number Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-22 Rong Zhang, Shu-Guang Guo
The -matrix of a graph was defined by Nikiforov in 2017 as , where , and are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of is called -index of . In this paper, we completely determine the extremal graphs with maximal -index among all graphs with size , domination number and no isolated vertices for .
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A note on the minimum size of matching-saturated graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-22 Xuechun Zhang, Hongliang Lu, Qinglin Yu
Let be positive integers and be a graph. A graph is called -saturated if is not a subgraph of but contain a copy of for every edge , where is the complement graph of . Let be the minimum number of edges over all -saturated graphs with order and denotes the family of -saturated graphs with edges and vertices. Let denote a matching of size and let denote a fractional matching of size . In this paper
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Matching extension and matching exclusion via the size or the spectral radius of graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-20 Shujing Miao, Shuchao Li, Wei Wei
A graph G is said to be k-extendable if every matching of size k in G can be extended to a perfect matching of G, where k is a positive integer. We say G is 1-excludable if for every edge e of G, there exists a perfect matching excluding e. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of G to guarantee that G is k-extendable. Then we determine a lower bound
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On the oriented achromatic number of graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-18 Pavan P.D., Éric Sopena
The oriented achromatic number of an oriented graph G⃗, denoted ψo(G⃗), is the largest n such that G⃗ admits a complete oriented n-colouring. If G is an undirected graph, the oriented achromatic number ψo(G) of G is the largest oriented achromatic number of its orientations. In this work we continue the study of oriented achromatic number of undirected graphs which was first initiated by Sopena (2014)
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On monotonicity in Maker–Breaker graph colouring games Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-19 Lawrence Hollom
In the Maker–Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a proper colouring at all times. Maker aims to colour the whole graph, and Breaker aims to make some vertex impossible to colour. We are interested in the following question, first asked by Zhu in 1999: if Maker wins with k colours
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The complexity of spanning tree problems involving graphical indices Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-19 Yanni Dong, Hajo Broersma, Yuhang Bai, Shenggui Zhang
We consider the computational complexity of spanning tree problems involving the graphical function-index. This index was recently introduced by Li and Peng as a unification of a long list of chemical and topological indices. We present a number of unified approaches to determine the NP-completeness and APX-completeness of maximum and minimum spanning tree problems involving this index. We give many
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Automating weight function generation in graph pebbling Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-19 Dominic Flocco, Jonad Pulaj, Carl Yerger
Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified root
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2-tone coloring of cactus graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-19 Allan Bickle
A 2-tone coloring of a graph assigns two distinct colors to each vertex with the restriction that adjacent vertices have no common colors, and vertices at distance two have at most one common color. The 2-tone chromatic number of a graph is the minimum number of colors in any 2-tone coloring. A cactus graph has every block a cycle or edge. We determine the 2-tone chromatic number of all cactus graphs
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Robust two-stage combinatorial optimization problems under discrete demand uncertainties and consistent selection constraints Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-20 Christina Büsing, Sabrina Schmitz
In this paper, we study a robust two-stage concept for combinatorial optimization problems under discrete demand uncertainty. Combinatorial optimization problems are based on a finite set of elements for which we decide whether they are part of a solution. We divide the elements into two types, the so-called fixed and free elements. In a first stage, we irrecoverably decide whether some fixed elements
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Irreducibility of recombination Markov chains in the triangular lattice Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-17 Sarah Cannon
In the United States, regions (such as states or counties) are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can have a profound effect on who is elected, and drawing the districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect when gerrymandering is occurring, but one algorithmic
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Feedback game on 3-chromatic Eulerian triangulations of surfaces Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-16 Akihiro Higashitani, Kazuki Kurimoto, Naoki Matsumoto
A new impartial game on a connected graph was introduced, called a feedback game, which is a variant of generalized geography. In this paper, we study the feedback game on 3-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every 3-chromatic Eulerian triangulation of a surface all of whose vertices have degree 0 modulo 4 is always fixed. Moreover, we also study
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Extremal digraphs for open neighbourhood location-domination and identifying codes Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-16 Florent Foucaud, Narges Ghareghani, Pouyeh Sharifani
A set S of vertices of a digraph D is called an open neighbourhood locating-dominating set if every vertex in D has an in-neighbour in S, and for every pair u,v of vertices of D, there is a vertex in S that is an in-neighbour of exactly one of u and v. The smallest size of an open neighbourhood locating-dominating set of a digraph D is denoted by γOL(D). We study the class of digraphs D whose only
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Random bipartite Ramsey numbers of long cycles Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-10 Meng Liu, Yusheng Li
For graphs G and H, let G⟶kH signify that any k-edge coloring of G contains a monochromatic H as a subgraph. Let G(K2(N),p) be random graph spaces with edge probability p, where K2(N) is the complete N×N bipartite graph. Let C2n be a cycle of length 2n and let k=2,3. It is shown for any ϵ>0, there exists T=T(ϵ)>0 such that if np>T, then Pr[G(K2((k+ϵ)n),p)⟶kC2n]→1 as n→∞, for which the proof relies
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Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-08 Wenhao Zhen, Dein Wong, Yuanshuai Zhang
For a connected graph G, we denote by mG(μ), c(G) and p(G) the eigenvalue multiplicity of μ in G, the cyclomatic number and the number of pendant vertices in G, respectively. In 2020, Wang et al. (2020) proved that mG(μ)≤2c(G)+p(G) for any μ∈R, the equality holds if and only if G is a cycle and μ is a multiple eigenvalue of the cycle. We find that when ∥μ∥≥2, the bound 2c(G)+p(G) is much rough. Thus
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On d-stable locally checkable problems parameterized by mim-width Discrete Appl. Math. (IF 1.1) Pub Date : 2024-01-03 Carolina Lucía Gonzalez, Felix Mann
In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of d-stability of a check function
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Safe sets and in-dominating sets in digraphs Discrete Appl. Math. (IF 1.1) Pub Date : 2023-12-29 Yandong Bai, Jørgen Bang-Jensen, Shinya Fujita, Hirotaka Ono, Anders Yeo
A non-empty subset S of the vertices of a digraph D is a safe set if (i) for every strongly connected component M of D−S, there exists a strongly connected component N of D[S] such that there exists an arc from M to N; and (ii) for every strongly connected component M of D−S and every strongly connected component N of D[S], we have |M|≤|N| whenever there exists an arc from M to N. In the case of acyclic
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Induced forests in some distance-regular graphs Discrete Appl. Math. (IF 1.1) Pub Date : 2023-12-26 Karen Gunderson, Karen Meagher, Joy Morris, Venkata Raghu Tej Pantangi
In this article, we study the order and structure of the largest induced forests in some families of graphs. First we prove a variation of the Delsarte–Hoffman ratio bound for cocliques that gives an upper bound on the order of the largest induced forest in a graph. Next we define a canonical induced forest to be a forest that is formed by adding a vertex to a coclique and give several examples of
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Complete bipartite graphs without small rainbow subgraphs Discrete Appl. Math. (IF 1.1) Pub Date : 2023-12-27 Zhiqiang Ma, Yaping Mao, Ingo Schiermeyer, Meiqin Wei
Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey number bgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow
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A notion of vertex equitability for proper labellings Discrete Appl. Math. (IF 1.1) Pub Date : 2023-12-27 Julien Bensmail
We introduce an equitable version of proper labellings of graphs, where the notion of equitability is with respect to the resulting vertex sums. That is, we are interested in k-labellings where, when computing the sums of labels incident to the vertices, we get a vertex-colouring that is not proper only, but also equitable. For a given graph G, we are interested in the parameter χ¯Σ(G), which is the
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Sprague–Grundy values and complexity for LCTR Discrete Appl. Math. (IF 1.1) Pub Date : 2023-12-23 Eric Gottlieb, Matjaž Krnc, Peter Muršič
Given an integer partition of n, we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of its Young diagram. We show that for both normal and misère play, the optimal strategy can consist mostly of mirroring the opponent’s moves. We also establish that both LCTR and Downright are domestic as well as returnable, and on the other hand neither