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Journal articles on the topic 'Graph classes'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

Kaladevi, V., R. Murugesan, and K. Pattabiraman. "First reformulated Zagreb indices of some classes of graphs." Carpathian Mathematical Publications 9, no. 2 (2018): 134–44. http://dx.doi.org/10.15330/cmp.9.2.134-144.

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A topological index of a graph is a parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. Graph operations plays a vital role to analyze the structure and properties of a large graph which is derived from the smaller graphs. The Zagreb indices are the important topological indices found to have the applications in Quantitative Structure Property Relationship(QSPR) and Quantitative Structure Activity Relationship(QSAR) studies as well. There are various study of different versions of Zagreb indices. One of the most important Zagreb indices is t

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2

Besirik, Ayse, and Elgin Kilic. "Domination integrity of some graph classes." RAIRO - Operations Research 53, no. 5 (2019): 1721–28. http://dx.doi.org/10.1051/ro/2018074.

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The stability of a communication network has a great importance in network design. There are several vulnerability measures used to determine the resistance of network to the disruption in this sense. Domination theory provides a model to measure the vulnerability of a graph network. A new vulnerability measure of domination integrity was introduced by Sundareswaran in his Ph.D. thesis (Parameters of vulnerability in graphs (2010)) and defined as DI(G) = min{|S| + m(G − S):S ∈ V(G)} where m(G − S) denotes the order of a largest component of graph G − S and S is a dominating set of G. The domin

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Jose, Bibin K. "Some New Classes of Open Distance-Pattern Uniform Graphs." International Journal of Combinatorics 2013 (July 24, 2013): 1–7. http://dx.doi.org/10.1155/2013/863439.

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Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this pape

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Pushpam, P. Roushini Leely, and D. Yokesh. "Differentials in certain classes of graphs." Tamkang Journal of Mathematics 41, no. 2 (2010): 129–38. http://dx.doi.org/10.5556/j.tkjm.41.2010.664.

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Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regula

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A. Antony mary, A., A. Amutha, and M. S. Franklin Thamil Selvi. "A Study on Slope Number of Certain Classes of Bipartite Graphs." International Journal of Engineering & Technology 7, no. 4.10 (2018): 440. http://dx.doi.org/10.14419/ijet.v7i4.10.21036.

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Graph drawing is the most important area of mathematics and computer science which combines methods from geometric graph theory and information visualization. Generally, graphs are represented to explore some intellectual ideas. Graph drawing is the familiar concept of graph theory. It has many quality measures and one among them is the slope number. Slope number problem is an optimization problem and is NP-hard to determine the slope number of any arbitrary graph. In the present paper, the investigation on slope number of bipartite graph is studied elaborately. Since the bipartite graphs crea

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Aref'ev, Roman D., John T. Baldwin та Marco Mazzucco. "Classification of δ-invariant amalgamation classes". Journal of Symbolic Logic 64, № 4 (1999): 1743–50. http://dx.doi.org/10.2307/2586809.

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Hrushovski's generalization of the Fraisse construction has provided a rich source of examples in model theory, model theoretic algebra and random graph theory. The construction assigns to a dimension function δ and a class K of finite (finitely generated) models a countable ‘generic’ structure. We investigate here some of the simplest possible cases of this construction. The class K will be a class of finite graphs; the dimension, δ(A), of a finite graph A will be the cardinality of A minus the number of edges of A. Finally and significantly we restrict to classes which are δ-invariant. A cla

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7

GEORGAKOPOULOS, AGELOS, and STEPHAN WAGNER. "Subcritical Graph Classes Containing All Planar Graphs." Combinatorics, Probability and Computing 27, no. 5 (2018): 763–73. http://dx.doi.org/10.1017/s0963548318000056.

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8

Chandrasekar, K. Raja, and S. Saravanakumar. "OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS." Ural Mathematical Journal 6, no. 2 (2020): 38. http://dx.doi.org/10.15826/umj.2020.2.004.

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Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

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GOLUMBIC, MARTIN CHARLES, and UDI ROTICS. "ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES." International Journal of Foundations of Computer Science 11, no. 03 (2000): 423–43. http://dx.doi.org/10.1142/s0129054100000260.

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Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a u

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10

Vignesh, R., J. Geetha, and K. Somasundaram. "Total Coloring Conjecture for Certain Classes of Graphs." Algorithms 11, no. 10 (2018): 161. http://dx.doi.org/10.3390/a11100161.

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A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

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11

LaGrange, John D. "Characterizations of Three Classes of Zero-Divisor Graphs." Canadian Mathematical Bulletin 55, no. 1 (2012): 127–37. http://dx.doi.org/10.4153/cmb-2011-107-3.

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AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

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12

Nath, Rajat Kanti, and Jutirekha Dutta. "Spectrum of commuting graphs of some classes of finite groups." MATEMATIKA 33, no. 1 (2017): 87. http://dx.doi.org/10.11113/matematika.v33.n1.812.

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In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group G is integral if G is not isomorphic to the symmetric group of degree 4 and the commuting graph of G is planar. Further, it is shown that the commuting graph of G is integral if its commuting graph is toroidal.

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13

Pattabiraman, K. "F-Indices and its coindices of some classes of graphs." Creative Mathematics and Informatics 26, no. 2 (2017): 201–10. http://dx.doi.org/10.37193/cmi.2017.02.09.

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In this paper, first we investigate the basic properties of the F-index and its coindex of graph. Next we obtain the exact expression of F-indices and its coindices for bridge graph, chain graph and transformation of graph. Using some of these results, we have obtained the value of these indices for some important classes of chemical graphs.

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14

Power, Stephen C., Igor A. Baburin, and Davide M. Proserpio. "Isotopy classes for 3-periodic net embeddings." Acta Crystallographica Section A Foundations and Advances 76, no. 3 (2020): 275–301. http://dx.doi.org/10.1107/s2053273320000625.

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Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmet

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15

Cicerone, Serafino, and Gabriele Di Stefano. "Graph classes between parity and distance-hereditary graphs." Discrete Applied Mathematics 95, no. 1-3 (1999): 197–216. http://dx.doi.org/10.1016/s0166-218x(99)00075-x.

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16

Argiroffo, Gabriela, Valeria Leoni, and Pablo Torres. "-domination for chordal graphs and related graph classes." Electronic Notes in Discrete Mathematics 44 (November 2013): 219–24. http://dx.doi.org/10.1016/j.endm.2013.10.034.

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17

Sritharan, R. "Graph modification problem for some classes of graphs." Journal of Discrete Algorithms 38-41 (May 2016): 32–37. http://dx.doi.org/10.1016/j.jda.2016.06.003.

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18

Jansrang, Monsikarn, and Sivaram K. Narayan. "Graph complement conjecture for classes of shadow graphs." Operators and Matrices, no. 2 (2021): 589–614. http://dx.doi.org/10.7153/oam-2021-15-40.

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19

Golumbic, Martin Charles, and Robert E. Jamison. "Rank-tolerance graph classes." Journal of Graph Theory 52, no. 4 (2006): 317–40. http://dx.doi.org/10.1002/jgt.20163.

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20

ADDARIO-BERRY, L., C. MCDIARMID, and B. REED. "Connectivity for Bridge-Addable Monotone Graph Classes." Combinatorics, Probability and Computing 21, no. 6 (2012): 803–15. http://dx.doi.org/10.1017/s0963548312000272.

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A classof labelled graphs isbridge-addableif, for all graphsGinand all verticesuandvin distinct connected components ofG, the graph obtained by adding an edge betweenuandvis also in; the classismonotoneif, for allG∈and all subgraphsHofG, we haveH∈. We show that for any bridge-addable, monotone classwhose elements have vertex set {1,. . .,n}, the probability that a graph chosen uniformly at random fromis connected is at least (1−on(1))e−½, whereon(1) → 0 asn→ ∞. This establishes the special case of the conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This r

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21

Chandoor, Susanth, and Sunny Joseph Kalayathankal. "Operations on covering numbers of certain graph classes." International Journal of Advanced Mathematical Sciences 4, no. 1 (2016): 1. http://dx.doi.org/10.14419/ijams.v4i1.5531.

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<p><span>The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In a similar way, the operations on the covering numbers of graphs with their complement are studied and with respect to this, new characterizations of certain graph classes have also been given in this paper.</span></p>

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CSIKVÁRI, PÉTER, and ZOLTÁN LÓRÁNT NAGY. "The Density Turán Problem." Combinatorics, Probability and Computing 21, no. 4 (2012): 531–53. http://dx.doi.org/10.1017/s0963548312000016.

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LetHbe a graph onnvertices and let the blow-up graphG[H] be defined as follows. We replace each vertexviofHby a clusterAiand connect some pairs of vertices ofAiandAjif (vi,vj) is an edge of the graphH. As usual, we define the edge density betweenAiandAjasWe study the following problem. Given densities γijfor each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graphG[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic toH,i.e., noHappears as a transversal inG[H]. We calldcrit(H) the maximal va

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23

Nguyen, Ngoc Tuy, Jörg Bornemann, and Van Bang Le. "Graph classes related to chordal graphs and chordal bipartite graphs." Electronic Notes in Discrete Mathematics 27 (October 2006): 73–74. http://dx.doi.org/10.1016/j.endm.2006.08.062.

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24

Domagalski, Rachel, and Sivaram Narayan. "Tree Cover Number and Maximum Semidefinite Nullity of Some Graph Classes." Electronic Journal of Linear Algebra 36, no. 36 (2020): 678–93. http://dx.doi.org/10.13001/ela.2020.5319.

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Let $G$ be a graph with a vertex set $V$ and an edge set $E$ consisting of unordered pairs of vertices. The tree cover number of $G$, denoted $\tau(G)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. In this paper, the tree cover number of a line graph $\tau(L(G))$ is shown to be equal to the path number $\pi(G)$ of $G$. Also, the tree cover numbers of shadow graphs, corona and Cartesian product of two graphs are found. The graph parameter $\tau(G)$ is related to another graph parameter $M_+(G)$, called the

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25

Fang, Wei, Wei-Hua Liu, Jia-Bao Liu, Fu-Yuan Chen, Zhen-Mu Hong, and Zheng-Jiang Xia. "Maximum Detour–Harary Index for Some Graph Classes." Symmetry 10, no. 11 (2018): 608. http://dx.doi.org/10.3390/sym10110608.

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The definition of a Detour–Harary index is ω H ( G ) = 1 2 ∑ u , v ∈ V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and l ( u , v | G ) is equal to the length of the longest path between vertices u and v. In this paper, we obtained the maximum Detour–Harary index about unicyclic graphs, bicyclic graphs, and cacti, respectively.

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BERNASCONI, NICLA, KONSTANTINOS PANAGIOTOU, and ANGELIKA STEGER. "The Degree Sequence of Random Graphs from Subcritical Classes." Combinatorics, Probability and Computing 18, no. 5 (2009): 647–81. http://dx.doi.org/10.1017/s0963548309990368.

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In this work we determine the expected number of vertices of degreek=k(n) in a graph withnvertices that is drawn uniformly at random from asubcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected values.

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27

Grohe, Martin, and Daniel Neuen. "Isomorphism, canonization, and definability for graphs of bounded rank width." Communications of the ACM 64, no. 5 (2021): 98–105. http://dx.doi.org/10.1145/3453943.

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We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with count

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Naduvath, Sudev. "Equitable coloring parameters of certain graph classes." Discrete Mathematics, Algorithms and Applications 10, no. 03 (2018): 1850040. http://dx.doi.org/10.1142/s1793830918500404.

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Coloring the vertices of a graph [Formula: see text] subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable [Formula: see text] can be defined as the color of a vertex chosen at random, with respect to the given type of coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. A proper coloring [Formula: see text] of a graph [Formula: see text], which assigns colors to the vertices of [Formula: see text] such that the numbers of vertices in any two color cla

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29

Zhou, Xiaoqing, Mustafa Habib, Tariq Javeed Zia, Asim Naseem, Anila Hanif, and Ansheng Ye. "Topological invariants for the line graphs of some classes of graphs." Open Chemistry 17, no. 1 (2019): 1483–90. http://dx.doi.org/10.1515/chem-2019-0154.

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AbstractGraph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomial

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30

Ashrafi, Ali, and Fatemeh Koorepazan-Moftakhar. "On normal graph of a finite group." Filomat 32, no. 11 (2018): 4047–59. http://dx.doi.org/10.2298/fil1811047a.

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Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A U B ? N. The aim of this paper is to study the normal graph of a finite group G. It is proved, among other things, that the groups with identical character table have isomorphic normal graphs and so this new graph associated to a group has good relationship by its group structure. The normal graphs of some classes of fin

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31

Pattabiraman, K., and P. Kandan. "Weighted PI index of corona product of graphs." Discrete Mathematics, Algorithms and Applications 06, no. 04 (2014): 1450055. http://dx.doi.org/10.1142/s1793830914500554.

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In this paper, we present exact formula for the weighted PI index of corona product of two connected graphs in terms of other graph invariants including the PI index, first Zagreb index and second Zagreb index. Then, we apply our result to compute the weighted PI indices of t-fold bristled graph, bottleneck graph, sunlet graph, star graph, fan graph, wheel graph and some classes of bridge graphs.

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Alekseev, V. E., R. Boliac, D. V. Korobitsyn, and V. V. Lozin. "NP-hard graph problems and boundary classes of graphs." Theoretical Computer Science 389, no. 1-2 (2007): 219–36. http://dx.doi.org/10.1016/j.tcs.2007.09.013.

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Bonomo, Flavia, and Jayme L. Szwarcfiter. "Characterization of classical graph classes by weighted clique graphs." Discrete Applied Mathematics 165 (March 2014): 83–95. http://dx.doi.org/10.1016/j.dam.2013.04.013.

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34

Keil, J. Mark, and Carl A. Gutwin. "Classes of graphs which approximate the complete euclidean graph." Discrete & Computational Geometry 7, no. 1 (1992): 13–28. http://dx.doi.org/10.1007/bf02187821.

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Águeda, Raquel, Nathann Cohen, Shinya Fujita, et al. "Safe sets in graphs: Graph classes and structural parameters." Journal of Combinatorial Optimization 36, no. 4 (2017): 1221–42. http://dx.doi.org/10.1007/s10878-017-0205-2.

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Chapuy, G., and G. Perarnau. "Local Convergence and Stability of Tight Bridge-addable Classes." Canadian Journal of Mathematics 72, no. 3 (2018): 563–601. http://dx.doi.org/10.4153/s0008414x18000020.

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AbstractA class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$-vertex graph from ${\mathcal{G}}$, then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.In this paper, we prove a form of uniqueness in this statement: if $

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Fornasiero, Federico, and Sudev Naduvath. "On J-colorability of certain derived graph classes." Acta Universitatis Sapientiae, Informatica 11, no. 2 (2019): 159–73. http://dx.doi.org/10.2478/ausi-2019-0011.

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Abstract A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every color class of G with respect to C. A maximal proper coloring of a graph G is a J-coloring of G such that every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-coloring of certain Mycielski-type graphs.

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Chua, Dhenmar Enriquez, Francis Joseph Hernandez Campeña, and Floresto Franco. "Efficient Zero Ring Labeling of Graphs." European Journal of Pure and Applied Mathematics 13, no. 3 (2020): 674–96. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3780.

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A zero ring is a ring in which the product of any two elements is zero, which is the additive identity. A zero ring labeling of a graph is an assignment of distinct elements of a zero ring to the vertices of the graph such that the sum of the labels of any two adjacent vertices is not the zero element in the ring. Given a zero ring labeling of a graph, if the cardinality of the set of distinct sums obtained from all adjacent vertices is equal to the maximum degree of the graph, then the zero ring labeling is efficient. In this paper, we showed the existence of an efficient zero ring labeling f

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CHEN, XIAOXUAN, JING YANG, XIANYA GENG, and LONG WANG. "SINGULARITY OF ORIENTED GRAPHS FROM SEVERAL CLASSES." Bulletin of the Australian Mathematical Society 102, no. 1 (2019): 7–14. http://dx.doi.org/10.1017/s0004972719001151.

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A digraph is called oriented if there is at most one arc between two distinct vertices. An oriented graph $D$ is nonsingular if its adjacency matrix $A(D)$ is nonsingular. We characterise all nonsingular oriented graphs from three classes: graphs in which cycles are vertex disjoint, graphs in which all cycles share exactly one common vertex and graphs formed by cycles sharing a common path. As a straightforward corollary, the singularity of oriented bicyclic graphs is determined.

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Genova, Daniela, Hendrik Jan Hoogeboom, and Nataša Jonoska. "Companions and an Essential Motion of a Reaction System." Fundamenta Informaticae 175, no. 1-4 (2020): 187–99. http://dx.doi.org/10.3233/fi-2020-1953.

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For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-conte

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Imran, Muhammad, Yasir Ali, Mehar Ali Malik, and Kiran Hasnat. "Chromatic spectrum of some classes of 2-regular bipartite colored graphs." Journal of Intelligent & Fuzzy Systems 41, no. 1 (2021): 1125–33. http://dx.doi.org/10.3233/jifs-210066.

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Chromatic spectrum of a colored graph G is a multiset of eigenvalues of colored adjacency matrix of G. The nullity of a disconnected graph is equal to sum of nullities of its components but we show that this result does not hold for colored graphs. In this paper, we investigate the chromatic spectrum of three different classes of 2-regular bipartite colored graphs. In these classes of graphs, it is proved that the nullity of G is not sum of nullities of components of G. We also highlight some important properties and conjectures to extend this problem to general graphs.

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Bień, Anna. "Gamma Graphs Of Some Special Classes Of Trees." Annales Mathematicae Silesianae 29, no. 1 (2015): 25–34. http://dx.doi.org/10.1515/amsil-2015-0003.

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AbstractA set S ⊂ V is a dominating set of a graph G = (V, E) if every vertex υ ∈ V which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if |S| = γ(G).Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second secti

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ERDŐS, PÉTER L., ISTVÁN MIKLÓS, and ZOLTÁN TOROCZKAI. "New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling." Combinatorics, Probability and Computing 27, no. 2 (2017): 186–207. http://dx.doi.org/10.1017/s0963548317000499.

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In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampli

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Borowiecki, Mieczysław, Piotr Borowiecki, Ewa Drgas-Burchardt, and Elżbieta Sidorowicz. "Graph classes generated by Mycielskians." Discussiones Mathematicae Graph Theory 40, no. 4 (2020): 1163. http://dx.doi.org/10.7151/dmgt.2345.

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Kawarabayashi, Ken-ichi, Michael D. Plummer, and Akira Saito. "On two equimatchable graph classes." Discrete Mathematics 266, no. 1-3 (2003): 263–74. http://dx.doi.org/10.1016/s0012-365x(02)00813-0.

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Kwak, Jin Ho, and Jaeun Lee. "Isomorphism Classes of Graph Bundles." Canadian Journal of Mathematics 42, no. 4 (1990): 747–61. http://dx.doi.org/10.4153/cjm-1990-039-3.

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AbstractRecently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

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Kratochvil, Jan, and Zsolt Tuza. "Intersection Dimensions of Graph Classes." Graphs and Combinatorics 10, no. 2-4 (1994): 159–68. http://dx.doi.org/10.1007/bf02986660.

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Balister, Paul, Béla Bollobás, and Stefanie Gerke. "Connectivity of addable graph classes." Journal of Combinatorial Theory, Series B 98, no. 3 (2008): 577–84. http://dx.doi.org/10.1016/j.jctb.2007.09.003.

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Belmonte, Rémy, Pinar Heggernes, Pim van ’t Hof, Arash Rafiey, and Reza Saei. "Graph classes and Ramsey numbers." Discrete Applied Mathematics 173 (August 2014): 16–27. http://dx.doi.org/10.1016/j.dam.2014.03.016.

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Golovach, Petr A., Pinar Heggernes, Dieter Kratsch, and Arash Rafiey. "Finding clubs in graph classes." Discrete Applied Mathematics 174 (September 2014): 57–65. http://dx.doi.org/10.1016/j.dam.2014.04.016.

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