Chordal graph properties. Where no reference is given, check equivalent classes or use the Java application. For an undirected graph G the k-th power G~ of G isthe graph with the same vertex s as t G where two vertices ar adjacent iff their distance isat most k in G. Minimal/maximal is with respect to the contents of ISGCI. We show that From Bayesian Networks to MNs Two perspectives Given a BN B how to represent the distribution PB as a parameterized MN OR Given graph G how to represent independencies in G using an undirected graph H Oct 26, 2025 · Chordality properties on graphs and minimal conceptual connections in semantic data models J. For instance, Chordal and Ptolemaic graphs have been characterized as convex geometries with respect to the monophonic convexity and geodesic convexity respectively [6]. A chordal graph is a simple graph in which chordal graph is an undirected graph with the property that every cycle of length greater than three has a chord (an edge between noncon-secutive vertices in the cycle). Also called rigid circuit graphs, Perfect Elimination Graphs, Triangulated Graphs, monotone transitive graphs. We derive a formula for the chromatic polynomial of a chordal or a triangulated graph in terms of its maximal cliques. For instance, being a perfect graph or being a chordal graph are hereditary properties. Chordal graphs are important in graph theory due to their structural properties and the fact that they allow Peter Bartlett. Oct 28, 2025 · A chordal graph is a simple graph in which every graph cycle of length four and greater has a cycle chord. Chordal graphs. 2 Chordal Graphs De nition 2. October 2003. A graph G is chordal if every cycle oflength a least four has a chord. 1 A comparability graph is an undirected graph in which it is possible to orient each edge such that the resultant graph (G=(V, U)) has the following properties Anti-symmetry: If edge u ! v exists, then v ! u does not. In particu-lar, it follows that for strongly chordal graphs and so-called power chordal graphs the injective chromatic number can be determined in polynomial time. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. The problem of interest is to May 31, 2024 · Abstract A graph class G G has the strong Erd {\H o}s--Hajnal property (SEH-property) if there is a constant c= c(G)> 0 c = c (G)> 0 such that for every member G G of G G, either G G or its complement has Km,m K m, m as a subgraph where m≥ ⌊c|V (G)|⌋ m ≥ ⌊ c | V (G) | ⌋. All chordal graphs have PEO, but no proof of this yet. In particular T can be chosen such that each node of T corresponds to a maximal clique of G and the subtrees T v consist of precisely those maximal cliques in G that contain v. Apr 28, 1997 · Abstract A graph is chordal or triangulated if it has no chordless cycle with four or more vertices. Golumbic, C. Aug 1, 2024 · A graph is chordal if it does not contain an induced cycle of length at least 4. Our next goal is to determine (asymptotically) optimal constants for the CEH-property with respect to the same graph classes as in Theorem 1. G. Abstract The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. A chord of a cycle C in a simple graph G is an edge in E(G) \ E(C) both of those ends lie on C (see Exercise 2. Only references for direct inclusions are given. The semitotal domination number, denoted by t2(G), is the minimum cardinality of a semitotal dominating set of G. The concept of H H -graphs, introduced by Biró, Hujter, and Tuza, naturally captures interval graphs, circular-arc graphs, and chordal graphs, among others. West 2000, p. The algorithm uses the metric properties of chordal graphs and Tarjan It was shown in [1] that semi-algebraic graphs satisfy the CEH-property, but this general and powerful result does not give particularly good bounds on the constant involved. (A chordal graph is sometimes called a triangulated graph in the literature. Also, the strength of the chords is not measured there A chordal graph is a type of graph where every cycle of length at least 4 contains an edge connecting two non-adjacent vertices. Introduction This paper introduces the class of chordal probe graphs, a new family of perfect graphs. 21 hours ago · This extends previous work, which was primarily focused on co-chordal graphs [AhmedMafiNamiq2025, Froberg2022, GitlerValencia2005, MoradiKiani2010], to a classification of all graphs with this property. [1] A graph property is monotone if every subgraph of a graph with property P also Perfect graphs, by de nition, have a nice structure, that graph searching seems to extract in a, often non-inexpensive, manner. In this paper weconsider LexBFS-orderings of chordal, distance-hereditary and HHD-free graphs (the graphs where each cycle oflength atleast five has two chords) with re-spect to heir powers. A perfect elimination ordering (see chordal ) of a graph G is a strong perfect elimination ordering if for all i<j, k<l: if v i v k , v i v l , v j v k are edges, then v j v l is an edge. For instance a 3x3 grid is a median graph (it has a unique median for every three vertices, a stronger version of your property 2 which does not require uniqueness) but it is not chordal bipartite (the outer 8-cycle has no chord). 55 1-18 (2024) Abstract Chordal graphs are those graphs which have chords for each cycle of the length > 3. By carefully exploiting the properties of this graph (in particular, its chordal completions), more e cient algorithms can be developed. In chordal graphs, the maximal cliques (the largest cliques that cannot be extended) play a key role in 5 days ago · I(G) has a linear resolution ⇐⇒ G is co-chordal, where a graph is said to be co-chordal if its complement is chordal. Although the notion of Alexander duality comes from the theory of simplicial complexes, we consider the equivalent algebraic definition in terms of Stanley–Reisner ideals. T is structure is hown to be optimal with regard tothe set ofclique trees. l Until recently, explicit use of the properties of chordal graphs or clique trees in sparse matrix computations was rarely needed. For strictly chordal graphs, we show new results about integer Laplacian eigenvalues which are directly related to particular vertex sets of the graph. Jun 9, 2011 · When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. [27]. We propose a novel decomposition of reduced clique graphs based on two operations: edge contraction and Sep 3, 2025 · We now proceed to study the vertex splittable property of Alexander dual of the t -path ideals of chordal graphs. While various parameterized Topics in Algorithmic Graph Theory - June 2021To save this book to your Kindle, first ensure coreplatform@cambridge. The following properties are equivalent: [4] G has a maximum neighborhood ordering. A newgreedy algorithm that generalizes both Maximal c rdinality Search (MCS) and Lexicographic B eadth first search is presented Properties of Chordal Graphs Simplicity: Chordal graphs are easier to study than many other types of graphs due to their structured nature. The structure of a system of polynomial equations can be described in terms of a graph. Graph Theory 2 1978 155--163 Sep 2, 2024 · The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. chordal graph is an undirected graph with the property that every cycle of length greater than three has a chord (an edge between noncon-secutive vertices in the cycle). We study algorithmic properties of the graph class Chordal ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of ll-in at most k. t. These graphs have numerous applications in various fields, including computer science, operations research, and combinatorial optimization. These are chordal. For large graphs/ networks, generally, the number of chords is less in number than the required number of chords to for a chordal graph. The center ofa graph isthe set ofvertices with minimum eccentricity. These graphs axe dual (in the sense 1. Here we introduce a generalization of chordal graphs, namely CSG k graphs. In a graph G = (V, E), the eccentricity e(v) of a vertex vis max{d(v,u) : u E V}. All trees are chordal. Chordal graphs have several useful properties. Step3:))Find)Maximal)Cliques • Maximum)cardinality)search:))gives)an)induced) graph)with)no)fill)edges)(assuming)chordal) graph)as)input,)which)we)have). That is, proper interval = clique graphs graphs of proper interval. In other words, a chordal graph is a graph possessing no chordless cycles of length four or greater (cf. Abstract We investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. Apr 28, 2009 · Abstract We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. A chordal graph is an undirected graph with the property that every cycle of length greater than three has a chord (an edge between nonconsecutive vertices in the cycle). These graphs are useful in solving complex graph problems, such as finding maximal cliques, as they can simplify the task. There are several related characterization of chordal graphs. The greedy aspect of the recognition algorithms of chordal graphs is tudied. We scratch the surface of this elegant research area by giving two examples: Lexicographic Breadth Search on Chordal Graphs, and Lexicographic Depth First Search on Cocomparability graphs. One of the other motivations we had for de ning perfect graphs was our hope that it would lead us to a \nice" characterizing property, similar to the one we had for bipartite graphs; does one exist for these \perfe t elimination orderin urns out, 2 Chordal Graphs Dec 1, 2012 · Many works related to dually chordal graphs, their cliques and neighborhoods were published by Brandstadt et al. Moreover, for chordal graphs in general, we show that the decision problem with a fixed number of colours is solvable in polynomial time. Also, the strength of the chords is not Abstract A graph class G has the strong Erdős–Hajnal property (SEH-property) if there is a constant c = c (G) > 0 such that for every member G of G, either G or its complement has K m,m as a subgraph where (Formula presented). The most important source classes include planar graphs [20, 18, 16], bipartite graphs [32], and degree-bounded graphs [19]. In this report, we focus on a class of graphs with particularly nice structure, namely chordal graphs. These notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We characterize K q -free Toeplitz graphs for an integer q ≥ 3 and give equivalent conditions for a Toeplitz graph G n 〈t 1, t 2, …, t k〉 with t 1 < ⋯ < t k and n ≥ t k − 1 + t k being chordal and equivalent conditions for a Toeplitz graph G n 〈t 1, t 2〉 being perfect. Then enter the ‘name’ part of your Kindle email address below. In this section, we define a chordal graph, give a tree-like construction of the (called a simplicial decomposition), and give two classifications of chordal graphs (in Corollary 9. A graph G is chordal if it is the intersection graph of subtrees of a tree T. Introduction graph is said to be chordal if every cycle with at least 4 vertices always contains a chord. In this comprehensive article, we will explore the theoretical foundations of chordal graphs, algorithms for working with them, and their real-world applications. Every monotone Oct 26, 2025 · Inclusions The map shows the inclusions between the current class and a fixed set of landmark classes. Chordal graphs have attracted interest in graph theory because several combinatorial optimization problems that are very difficult in general turn out to be easy for chordal graphs and solvable by simple greedy algorithms A graph G is chordal if every minimal cutset in every induced subgraph of G is a clique. Apr 1, 2021 · Request PDF | Fuzzy Chordal Graphs and Its Properties | Chordal graphs are those graphs which have chords for each cycle of the length > 3. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. , the intersection graphs of maximal cliques of chordal graphs. As one may expect, with special properties imposed on input graphs, the problems become easier, and some of them may not remain NP-hard. Comput. May 27, 2025 · In this article, we will explore the world of chordal graphs, their properties, and their significance in different areas. The decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar graphs. On the other hand, a strengthening of SEH-property which we call the colorful Apr 19, 2025 · We prove that chordal graphs are chromatic-choosable and present a decomposition theorem to help estimate the choice numbers for certain classes of chordless and non-chordal graphs. • Other)approaches)exist. This class is fixed under the clique operator. May 27, 2025 · Mastering Chordal Graphs: Theory and Practice Dive into the world of chordal graphs and discover their importance in graph theory, computer science, and beyond. 437). A graph is locally chordal if, locally at each vertex, it is chordal, i. _ Until recently, explicit use of the properties of chordal graphs or clique trees in sparse matrix computations was rarely needed A graph property P is hereditary if every induced subgraph of a graph with property P also has property P. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the Clearly, the family of chordal graphs is a hereditary property with an infinite set of minimal obstructions, but chordal graphs are characterized as those graphs that admit a B1-free acyclic orientation [16], i. 22 and Theorem 9. Dually chordal graphs are the clique graphs of chordal graphs, [3] i. In hefirst part of his paper, anew structure for chordal graph is introduced, nam the ly clique graph. Jun 12, 2025 · A look into the relationships and characteristics of circular-arc and chordal graphs. A graph is edge-triangular if every edge is in at least one triangle. As a corollary we obtain a way to write down an explicit formula for the chromatic polynomial for an arbitrary power of a graph which belongs to any given class of chordal graphs that are closed under taking powers. A graph G is called H - free for some graph H if G does not contain an induced subgraph isomorphic to H. Sep 10, 2016 · Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. 3. In addition to extending many classic characterizations of chordal graphs, we show that locally chordal graphs are precisely those graphs G which can be decomposed into cliques arranged in the shape of a high-girth graph. An implementation of the algorithm is presented For an interval graph, its PQ-tree represents all its possible models and can be taken as a cannonical representation of the graph (for example for graph isomorphism) But even path graphs are isomorphism complete. A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree. From a collection of subtrees of a tree, one can define a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees Jul 23, 2025 · This property makes chordal graphs simpler to work with in several computational and theoretical settings. Chordal graphs have attracted interest in graph theory because several combinatorial optimization problems that are very difficult in general turn out to be easy for chordal graphs and solvable by simple greedy algorithms Chordal Graphs: Theory and Algorithms Chordal graphs Chordal graph : Every cycle of four or more vertices has a chord in it, i. in [6] proved that the problem can be solved in polynomial time for bounded MIM-width graphs which includes many well known Oct 26, 2025 · A graph is weakly chordal if every connected induced subgraph H that is not a complete graph, contains a 2-pair. A graph is strictly chordal if it is chordal and its clique hypergraph is a strict hypertree. Sep 30, 2004 · 1. Sciences 33 1986 179--202 [457] M. For instance, being a bipartite graph or being a triangle-free graph is monotone. As a corollary, a characterization of minimally t -tough interval graphs is obtained for t ≤ 1 / 2. For graph-theoretical terminology, the reader is referred to [4]. In particular, optimal coloring of a chordal graph G= (V;E) can be done in O(jEj+ jVj) time. The first few are Chordal Graphs Note. Abstract. Clique: A clique is a group of vertices that are all connected to each other. Chordal graphs are well known for their combinatorial and algorithmic properties. However, it will be evident that the class of chordal graphs does not satisfy . Syst. Theoretical Foundations of Chordal Graphs Definition and Properties of A chordal graph, also known as aChordal Graphs A chordal graph, also known as a "perfectly triangulated graph," is a special type of graph where every cycle of length greater than 3 contains a chord. For example, chordal graphs are mentioned in a single exercise in George and Liu [16]. there is an edge between two non consecutive vertices of the cycle. A chord is an edge that connects two non-adjacent vertices of a cycle. Note In this thesis, we focus on generalized colouring problems in chordal graphs. Jun 16, 2025 · Chordal graphs, a subclass of perfect graphs, have been a subject of interest in graph theory and algorithm design due to their unique structural properties. November 10, 2020 A graph G is chordal if it does not contain any induced cycle of length at least four; i. AI generated definition based on: Constraint Processing, 2003 As an application of the above theorem, we combinatorially characterize all Cohen-Macaulay 𝑡 t italic_t -connected ideals of chordal graphs, which ensures that the Cohen-Macaulay property of such ideals does not depend on the characteristic of the base field. For a chordal graph, a perfect elimination ordering is a perfect ordering: the number of the color used for any vertex is the The same question can be asked for other pair of source and object graph classes. Graphs with a PEO. e. Definition and Basic Properties of Chordal Graphs A chordal graph is a graph in which every cycle of length greater than 3 has a chord. Graphs are made of points called vertices connected by lines called 2. The above for-mulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs. A graph is weakly chordal if it is (anti-hole,hole)-free . There is a spanning tree T of G such that any maximal clique of G induces a subtree in T. We propose a dynamic algorithm that supports operations of deleting and/or inserting edges or vertices for weakly chordal graphs. Find out more about saving to your Kindle. , any ( 4)-cycle in G has a chord (an edge between non-consecutive vertices of the cycle). The algorithms and properties of chordal graphs do not apply to such cases. In this article, we will explore the role of chordal graphs in algorithm design, focusing on their Feb 24, 2021 · Chordal graphs are those graphs which have chords for each cycle of the length > 3. Feb 5, 2023 · A graph class G has the strong Erdős-Hajnal property (SEH-property) if there is a constant c = c(G)> 0 such that for every member G of G, either G or its complement has Km,m as a subgraph where m ≥⌊c|V(G)|⌋. g. We then study the chordal probe graphs which are also weakly chordal, a subfamily which includes both A graph G is chordal if no cycle of length greater than 3 is induced, that is, chordal = Forbv(holes). We prove that the class of chordal graphs satisfy SEH-property with constant c= 2/9 c = 2 / 9. Feb 1, 2024 · We review results concerning general graphs showing that the number of universal vertices and the degrees of twins provide integer Laplacian eigenvalues and their respective multiplicities. On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal Oct 26, 2025 · A proper interval graph is an interval graph that has an intersection model in which no interval properly contains another. We first survey results about colouring problems, ho-momorphism problems and partitioning problems in general graphs, as well as, describe useful graph decomposition techniques used in efficient algorithms for these problems. We prove that the class of chordal graphs satisfy SEH-property with constant c = 2/9. We characterize the minimally t -tough chordal graphs for all t ≤ 1 / 2. The numbers of simple chordal graphs on n=1, 2, nodes are 1, 2, 4, 10, 27, 94, 393, (OEIS A048193). Of the many ways to represent a chordal graph, a particularly useful and compact representation is provided by clique trees [24,45]. C. composed of cliques glued along a tree. The maximal clique Jul 23, 2025 · In mathematical graph theory, a chordal graph is one in which all cycles of four or more vertices have an edge called a chord. Many graph properties are well-behaved with respect to certain natural partial orders or preorders defined on graphs: A graph property P is hereditary if every induced subgraph of a graph with property P also has property P. We discover that a number of fundamental intractable optimiza-tion problems being parameterized by k admit subexponential algorithms on graphs from Chordal ke. We present analgorithm whichomputes inlinear time a central vertex of a chordal graph. Jan 15, 2025 · Graph convexity have been studied in many context, some studies are devoted to deter-mine if a graph convexity is a convex geometry. To check relations other than inclusion (e. Galby et al. An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees. The struc-ture of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. The main results can be summa-rized as follows: the class of maximal outerplanar graphs is precisely the intersection of any of the two following classes: the chordal graphs, the path-neighborhood graphs, the edge-triangular graphs having a tree as triangle graph. We prove a number of basic properties of chordal probe graphs, and give a result on the enhancement of a graph which generalizes a previous result of Zhang [15] on interval probe graphs. Moreover, we update the changes on the pa-rameters tree-width and tree-length for this class when the operations occurs. By definition, those graphs are not chordal. It is weil known that chordal graphs model the sparsity struc ture of the eholesky factor of a sparse positive definite matrix [40J. a graph Dec 7, 2012 · We re-consider perfect elimination digraphs, that were introduced by Haskins and Rose in 1973, and view these graphs as directed analogues of chordal graphs. Chordal graphs, also called triangulated graphs, are impor-tant in algorithmic graph theory. Graph theory : Treewidth (resp. In particular, for quadratic square-free monomial ideals, the property of having a linear resolution depends only on the combinatorial structure of the associated graph and not on the characteristic of the base field. Jan 2, 2025 · As a consequence of the above results, we completely characterize when the t -connected ideal of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. Peter Bartlett. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. Algorithms. Sep 1, 2006 · In spite of many rich structural properties of chordal bipartite graphs, it seems challenging to find an intersection graph characterization of this class of graphs, cf. Tuesday, June 02, 2020 (Lecture 7): Interval graphs and chordal graphs Interval graphs. 225; Gross and Yellen 2006, p. a graph G is chordal if and only if it admits an acyclic orientation G′ such that B1 is not an induced subdigraph of G′. We will undertake a similar study by considering minimal vertex separators and their properties Chordal graphs Chordal Graph: an undirected graph G whose minimum cycle contains 3 verticies. Greedy-coloring gives optimum coloring and maximum clique. A chord is defined as an edge that connects two non-adjacent vertices within the cycle. Until recently, explicit use of the properties of chordal graphs or clique trces in sparse matrix computations It is well known that chordal graphs model the sparsity structure of the Cholesky factor of a sparse positive definite matrix [39]. We will undertake a similar study by considering minimal vertex separators and their properties instead. interval graph). We find a necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex and Nov 1, 2022 · Abstract In this paper, we study structural properties of Toeplitz graphs. Recently inseveral papers ([10],[22],[42]) independently graphs with maximum neigh-bourhood orderings were characterized and turned out to be algorithmicany useful. Oct 19, 2025 · Tree-layout based graph classes: proper chordal graphs International Symposioum on Theoretical Aspects of Computer Science STACS 2024, LIPIcs 289 No. l Until recently, explicit use of the properties of chordal A graph G is said to be perfectly orderable if there is a sequence of its vertices with the property that, for any induced subgraph of G, a greedy coloring algorithm that colors the vertices in the induced sequence ordering is guaranteed to produce an optimal coloring. ) A chordal completion of a graph G is a chordal graph with the same vertex set as G which contains all edges of G. org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Our study of generalized colouring problems focuses on those problems in which the sets Pi are characterized by a Chordal graphs have several useful properties. (1998) [1] and Gutierrez (1996) [6]. [1] A graph property is monotone if every subgraph of a graph with property P also has property P. As a consequence of the above results, we completely characterize when the t-connected ideal of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. Then we compute the edge clique Abstract. Informally, a CSG 0 graph is a complete graph, and for ks > 0, the class of CSG k graphs is defined inductively in a such manner that CSG fore (agai ine graph of a bipartite graph. Definition. r. On the other hand No. 1 Comparability Graph De nition 3. Several structural properties of chordal graphs that are crucial for algorithmic applications carry over to the directed setting, including notions like simplicial vertices, perfect elimination orderings, and vertex layouts. In the case of mixed graphs, we can have edges (without direction Chordal graphs A graph is a chordal graph if and only if Every cycle longer than three vertices has a chord, another edge connecting two vertices of the cycle or It has an elimination ordering: an ordering of the vertices where the later neighbors of every vertex form a clique Dec 1, 2012 · Many works related to dually chordal graphs, their cliques and neighborhoods were published by Brandstädt et al. The closed neighborhood hypergraph N (G) of G is a hypertree. • We)proved)lastDme)thatevery)maximal)clique) in)the)induced)graph)equates)to)the)scope)of) an)intermediate)factor)from)VE. Properties and graphs that are not interval graphs. This structure allows for efficient algorithms to work with them. A chordal graph is a special type of undirected simple graph. disjointness) use the Java application, as well. For large graphs/ networks, generally, the number of Oct 26, 2025 · The clique hypergraph $\cal {H} (G)$ of a graph $G= (V,E)$ is the hypergraph on $V$ with the collection of all maximal cliques of $G$ as its hyperedges. One of the most central structured representations used for network simplification is chordal or triangulated graphs. C D L Due to their strong combinatorial properties, chordal graphs are one of the most extensively studied graph classes in graph theory and discrete optimization. The decomposition in the shape of a high-girth Oct 26, 2025 · A graph is strongly chordal if it is chordal and every cycle of even length at least 6 has an odd chord. 2 A Chordal Graph is a graph that does not contain an induced cycle of length greater than 4. Jan 1, 1991 · Clique trees and chordal graphs have carved out a niche for themselves in recent work on sparse matrix algorithms, due primarily to research questions associated with advanced computer architectures. This paper gives aunified framework f characterizations ~ of those graphs interms ofneighbourhood and clique hype~graphs which have the Helly property andwhose line graph ischordal. A graph is chordal if and only if every induced cycle of length 4 or more is such that any two non-adjacent vertices of the cycle are connected as a chord or edge. In this paper we generalise the defini-tion of chordal graphs to the class of directed graphs. Of the many ways to represent a chordal graph, a particularly useful and compact representation is pro vided by clique trees [24, 46J. F. 19). Introduction. Goss Perfect elimination and chordal bipartite graphs J. Greedy-algorithm for independent set on Markov property for undirected graphs We say ( ) satisfy the global Markov property (G) w. 2. Problems such as minimum coloring, maximum clique, maximum independent set and minimum covering by cliques, which are NP-complete in general, can be solved in polynomial time for chordal graphs [11]. Several structural properties of chordal graphs that are crucial for algorithmic applications carry over to the directed setting, including notions like simplicial ver-tices, perfect elimination We consider the problem of maintaining of graph properties that arise in its tree decomposition. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. Maximum)Cardinality)Search for)VE Apr 1, 2021 · Abstract In this paper, structural properties of chordal graphs are studied, establishing a relationship between these structures and integer Laplacian eigenvalues. 23). pathwidth) are very important graph parameters that measure distance from a chordal graph (resp. glndqjdmswlml6b9diuo2oma4eggjywei7log1s1mw8p4v