Chordal graph properties. An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees. Chordal graphs are well known for their combinatorial and algorithmic properties. 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. Tuesday, June 02, 2020 (Lecture 7): Interval graphs and chordal graphs Interval graphs. The structure of a system of polynomial equations can be described in terms of a graph. • Other)approaches)exist. Peter Bartlett. G. A chord is defined as an edge that connects two non-adjacent vertices within the cycle. [1] A graph property is monotone if every subgraph of a graph with property P also has property P. Goss Perfect elimination and chordal bipartite graphs J. The greedy aspect of the recognition algorithms of chordal graphs is tudied. 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. 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. However, it will be evident that the class of chordal graphs does not satisfy . t. Abstract. For large graphs/ networks, generally, the number of chords is less in number than the required number of chords to for a chordal graph. 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. 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. 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. Here we introduce a generalization of chordal graphs, namely CSG k graphs. There is a spanning tree T of G such that any maximal clique of G induces a subtree in T. 437). 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. There are several related characterization of chordal graphs. We will undertake a similar study by considering minimal vertex separators and their properties instead. Aug 1, 2024 · A graph is chordal if it does not contain an induced cycle of length at least 4. 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. In this paper we generalise the defini-tion of chordal graphs to the class of directed graphs. 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. Sciences 33 1986 179--202 [457] M. 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). In this comprehensive article, we will explore the theoretical foundations of chordal graphs, algorithms for working with them, and their real-world applications. Then enter the ‘name’ part of your Kindle email address below. 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. Introduction This paper introduces the class of chordal probe graphs, a new family of perfect graphs. Graph theory : Treewidth (resp. 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. These notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models. e. r. 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. For example, chordal graphs are mentioned in a single exercise in George and Liu [16]. 2 Chordal Graphs De nition 2. 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. The most important source classes include planar graphs [20, 18, 16], bipartite graphs [32], and degree-bounded graphs [19]. disjointness) use the Java application, as well. Chordal graphs have several useful properties. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. 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. By definition, those graphs are not chordal. 22 and Theorem 9. Algorithms. 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. 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. A graph G is called H - free for some graph H if G does not contain an induced subgraph isomorphic to H. 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. 2 A Chordal Graph is a graph that does not contain an induced cycle of length greater than 4. Syst. 3. 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. 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. 55 1-18 (2024) Abstract Chordal graphs are those graphs which have chords for each cycle of the length > 3. The semitotal domination number, denoted by t2(G), is the minimum cardinality of a semitotal dominating set of G. For instance, being a bipartite graph or being a triangle-free graph is monotone. Minimal/maximal is with respect to the contents of ISGCI. 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. Greedy-coloring gives optimum coloring and maximum clique. A graph G is chordal if every cycle oflength a least four has a chord. 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′. 19). 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). May 27, 2025 · In this article, we will explore the world of chordal graphs, their properties, and their significance in different areas. Chordal graphs are important in graph theory due to their structural properties and the fact that they allow Peter Bartlett. Comput. 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. Dually chordal graphs are the clique graphs of chordal graphs, [3] i. A graph is locally chordal if, locally at each vertex, it is chordal, i. Introduction graph is said to be chordal if every cycle with at least 4 vertices always contains a chord. 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. 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. 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. For instance, Chordal and Ptolemaic graphs have been characterized as convex geometries with respect to the monophonic convexity and geodesic convexity respectively [6]. Our next goal is to determine (asymptotically) optimal constants for the CEH-property with respect to the same graph classes as in Theorem 1. Graphs with a PEO. 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. C. 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. 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. All chordal graphs have PEO, but no proof of this yet. 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. We prove that the class of chordal graphs satisfy SEH-property with constant c= 2/9 c = 2 / 9. 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. In this report, we focus on a class of graphs with particularly nice structure, namely chordal graphs. Note In this thesis, we focus on generalized colouring problems in chordal graphs. By carefully exploiting the properties of this graph (in particular, its chordal completions), more e cient algorithms can be developed. That is, proper interval = clique graphs graphs of proper interval. 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. On the other hand No. 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. 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. As a corollary, a characterization of minimally t -tough interval graphs is obtained for t ≤ 1 / 2. These graphs axe dual (in the sense 1. Oct 28, 2025 · A chordal graph is a simple graph in which every graph cycle of length four and greater has a cycle chord. _ 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. 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. 1 Comparability Graph De nition 3. The algorithms and properties of chordal graphs do not apply to such cases. Moreover, we update the changes on the pa-rameters tree-width and tree-length for this class when the operations occurs. A graph is strictly chordal if it is chordal and its clique hypergraph is a strict hypertree. The decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar 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. Find out more about saving to your Kindle. Chordal graphs. Introduction. One of the most central structured representations used for network simplification is chordal or triangulated graphs. The closed neighborhood hypergraph N (G) of G is a hypertree. 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. 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. This class is fixed under the clique operator. , any ( 4)-cycle in G has a chord (an edge between non-consecutive vertices of the cycle). The numbers of simple chordal graphs on n=1, 2, nodes are 1, 2, 4, 10, 27, 94, 393, (OEIS A048193). 2. 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. , the intersection graphs of maximal cliques of chordal graphs. 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. Greedy-algorithm for independent set on Markov property for undirected graphs We say ( ) satisfy the global Markov property (G) w. The first few are Chordal Graphs Note. 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. Also called rigid circuit graphs, Perfect Elimination Graphs, Triangulated Graphs, monotone transitive graphs. 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). Galby et al. 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)|⌋. 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. We prove that the class of chordal graphs satisfy SEH-property with constant c = 2/9. Sep 30, 2004 · 1. 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. 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. 23). composed of cliques glued along a tree. T is structure is hown to be optimal with regard tothe set ofclique trees. All trees are chordal. November 10, 2020 A graph G is chordal if it does not contain any induced cycle of length at least four; i. 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. West 2000, p. Sep 10, 2016 · Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. We propose a dynamic algorithm that supports operations of deleting and/or inserting edges or vertices for weakly chordal 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. 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). 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. Apr 28, 1997 · Abstract A graph is chordal or triangulated if it has no chordless cycle with four or more vertices. Recently inseveral papers ([10],[22],[42]) independently graphs with maximum neigh-bourhood orderings were characterized and turned out to be algorithmicany useful. Of the many ways to represent a chordal graph, a particularly useful and compact representation is provided by clique trees [24,45]. [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. Graphs are made of points called vertices connected by lines called 2. 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) | ⌋. In particular, optimal coloring of a chordal graph G= (V;E) can be done in O(jEj+ jVj) time. These graphs have numerous applications in various fields, including computer science, operations research, and combinatorial optimization. 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. interval graph). 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. Moreover, for chordal graphs in general, we show that the decision problem with a fixed number of colours is solvable in polynomial time. 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 chordal if it is the intersection graph of subtrees of a tree T. Then we compute the edge clique Abstract. 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]. 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. F. We characterize the minimally t -tough chordal graphs for all t ≤ 1 / 2. (1998) [1] and Gutierrez (1996) [6]. 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. 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. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. The concept of H H -graphs, introduced by Biró, Hujter, and Tuza, naturally captures interval graphs, circular-arc graphs, and chordal graphs, among others. Every monotone Oct 26, 2025 · Inclusions The map shows the inclusions between the current class and a fixed set of landmark classes. The following properties are equivalent: [4] G has a maximum neighborhood ordering. 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. Chordal graphs, also called triangulated graphs, are impor-tant in algorithmic graph theory. 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. A graph is edge-triangular if every edge is in at least one triangle. 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. 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. 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. In hefirst part of his paper, anew structure for chordal graph is introduced, nam the ly clique graph. Properties and graphs that are not interval graphs. 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). 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. The center ofa graph isthe set ofvertices with minimum eccentricity. 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. We present analgorithm whichomputes inlinear time a central vertex of a chordal graph. 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. 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. g. 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. October 2003. A chord is an edge that connects two non-adjacent vertices of a cycle. It is weil known that chordal graphs model the sparsity struc ture of the eholesky factor of a sparse positive definite matrix [40J. A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree. • We)proved)lastDme)thatevery)maximal)clique) in)the)induced)graph)equates)to)the)scope)of) an)intermediate)factor)from)VE. In other words, a chordal graph is a graph possessing no chordless cycles of length four or greater (cf. there is an edge between two non consecutive vertices of the cycle. Definition. 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. For strictly chordal graphs, we show new results about integer Laplacian eigenvalues which are directly related to particular vertex sets of the graph. 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. Of the many ways to represent a chordal graph, a particularly useful and compact representation is pro vided by clique trees [24, 46J. 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. These graphs are useful in solving complex graph problems, such as finding maximal cliques, as they can simplify the task. Clique: A clique is a group of vertices that are all connected to each other. 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. pathwidth) are very important graph parameters that measure distance from a chordal graph (resp. Golumbic, C. 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. A chordal graph is a special type of undirected simple graph. 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]. 225; Gross and Yellen 2006, p. Only references for direct inclusions are given. ) A chordal completion of a graph G is a chordal graph with the same vertex set as G which contains all edges of G. [27]. 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. 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. These are chordal. We derive a formula for the chromatic polynomial of a chordal or a triangulated graph in terms of its maximal cliques. 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. 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 graphs or clique trees in sparse matrix computations was rarely needed. For graph-theoretical terminology, the reader is referred to [4]. (A chordal graph is sometimes called a triangulated graph in the literature. Where no reference is given, check equivalent classes or use the Java application. A graph is weakly chordal if it is (anti-hole,hole)-free . To check relations other than inclusion (e. While various parameterized Topics in Algorithmic Graph Theory - June 2021To save this book to your Kindle, first ensure coreplatform@cambridge. In a graph G = (V, E), the eccentricity e(v) of a vertex vis max{d(v,u) : u E V}. We discover that a number of fundamental intractable optimiza-tion problems being parameterized by k admit subexponential algorithms on graphs from Chordal ke. 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. 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. Jun 12, 2025 · A look into the relationships and characteristics of circular-arc and chordal graphs. This structure allows for efficient algorithms to work with them. 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). 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. Step3:))Find)Maximal)Cliques • Maximum)cardinality)search:))gives)an)induced) graph)with)no)fill)edges)(assuming)chordal) graph)as)input,)which)we)have). For instance, being a perfect graph or being a chordal graph are hereditary properties. 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. 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. Abstract We investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. 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. 5au vdo prjw nxlzf 4fh gruua fl8qy iui uzg gvcr