Tree-Structured Graphs and Hypergraphs

Transcription

1 Tree-Structured Graphs and Hypergraphs by Andreas Brandstädt, University of Rostock, Germany, and Feodor F. Dragan, Kent State University, Ohio, U.S.A. Contents 1 Introduction: Graphs With Tree Structure, Related Graph Classes and Algorithmic Implications 5 2 Basic Notions and Results Basic Graph Notions and Results Line graphs Transitively orientable graphs Some Basic Hypergraph Notions and Properties Chordal Graphs, Subclasses and Variants Chordal Graphs Structural Properties of Chordal Graphs Algorithmic Problems for Chordal Graphs Convexity in Chordal Graphs and Powers of Chordal Graphs Split Graphs Interval Graphs Characterizations of Interval Graphs Unit Interval Graphs Generalizations and Variants of Interval Graphs and Related Graph Classes Chords and Cycles α-acyclic Hypergraphs and Their Duals A Motivation from Relational Database Theory Hypergraph Parameters: Coloring, Packing and Covering Hypergraph 2-Coloring Matchings and Transversals: The Kőnig Property Tree Structure of α-acyclic Hypergraphs

2 4.4 Join Trees as Maximum Spanning Trees Graham s Algorithm, Running Intersection Property and Other Desirable Properties Equivalent to α-acyclicity Dually Chordal Graphs Dually Chordal Graphs, Maximum Neighborhood Orderings and Hypertrees Bipartite Graphs, Hypertrees and Maximum Neighborhood Orderings Further Matrix Notions Totally Balanced Hypergraphs and Matrices Totally Balanced Hypergraphs Versus β-acyclic Hypergraphs Totally Balanced Matrices Strongly Chordal and Chordal Bipartite Graphs Strongly Chordal Graphs Elimination Orderings of Strongly Chordal Graphs Γ-Free Matrices and Strongly Chordal Graphs Strongly Chordal Graphs as Sun-Free Chordal Graphs Chordal Bipartite Graphs Tree Structure of Chordal Bipartite Graphs Relationship between Strongly Chordal and Chordal Bipartite Graphs Edge Elimination Orderings of Chordal Bipartite Graphs Recognition and Applications of Chordal Bipartite Graphs Distance-Hereditary Graphs, Subclasses and γ-acyclicity γ-acyclicity and Berge-Acyclicity Cographs Cograph characterization Tree Structure and Optimization on Cographs Distance-Hereditary Graphs Minimum Cardinality Steiner Tree Problem in Distance-Hereditary Graphs Further Important Subclasses of Distance-Hereditary Graphs Ptolemaic Graphs and Bipartite Distance-Hereditary Graphs

3 8.5.2 Block Graphs Tree Structure Decomposition of Graphs Modular Decomposition of Graphs Basic Module Properties Modular Decomposition Theorem Prime extensions of subgraphs in prime graphs Graph classes with simple prime graphs P 4 -Sparse Graphs Structure of co-chair-free chordal graphs Structure of bull-free chordal graphs Split Decomposition Homogeneous Decomposition Clique Separator Decomposition Treewidth and Clique-Width of Graphs Treewidth Clique-Width of Graphs Clique-Width of H-Free Bipartite Graphs Clique-Width of H-Free Chordal Graphs Clique-Width of (H 1,...,H k )-Free Graphs Power-Bounded Clique-Width of Graph Classes NLC-Width and Rank-Width of Graphs Leaf Powers Basic notions and results Basic facts on leaf powers Leaf powers are strongly chordal Leaf powers and fixed tolerance NeST graphs are the same class Leaf rank and unit interval graphs The inclusion structure of k-leaf power classes Characterizations of 3-leaf powers, 4-leaf powers and distance-hereditary 5-leaf powers Basic leaf powers and substitution of cliques

4 Characterizations of 3-leaf powers New characterizations of squares of trees Structure of (basic) 4-leaf powers Structure of (basic) distance-hereditary 5-leaf powers Variants and generalizations of leaf powers On (k,l)-leaf powers Exact leaf powers Simplicial powers of graphs Pairwise compatibility graphs and leaf powers Complexity of Some Problems on Tree-Structured Graph Classes Recognition Independent Set and Chromatic Number Hamiltonian Circuit Dominating Set and Steiner Tree Maximum Induced Matchings MIM for hypergraphs Exact Cover, Efficient Domination, and Efficient Edge Domination Exact Cover Efficient Domination Efficient Edge Domination Graph Isomorphism Metric Tree-Like Structures in Graphs Tree-Breadth, Tree-Length and Tree-Stretch of Graphs Hyperbolicity of Graphs and Embedding Into Trees

5 1 Introduction: Graphs With Tree Structure, Related Graph Classes and Algorithmic Implications The aim of this book is to present various aspects of tree structure in graphs and hypergraphs and its algorithmic implications together with some important graph classes having nice and useful tree structure. In particular, we describe the hypergraph background and the tree structure of chordal graphs and some graph classes which are closely related to chordal graphs such as chordal bipartite graphs, dually chordal graphs and strongly chordal graphs as well as important subclasses. A graph is chordal if it does not contain chordless cycles with at least four vertices as induced subgraphs. The study of chordal graphs goes back to [259], and the many aspects of chordal graphs are described in surveys and monographs such as [47, 93, 236, 359] and others. The interest in chordal graphs and related classes comes from applications in computer science, in particular, relational database schemes [203, 204], matrix analysis, models in biology, statistics, and others. The tree structure of chordal graphs is closely related to the famous concept of treewidth introduced by Robertson andseymour [398]; this concept appearsalso under thename ofpartial k-trees in [8, 10] (see e.g., [49]). Treewidth plays a central role in algorithmic and complexity aspects on graphs. Chordal graphs appear in the literature under various names such as triangulated graphs (Chapter 4 of[236]), rigid-circuit graphs, perfect elimination graphs and others. Most of the applications are due to the tree structure of chordal graphs which can be described in terms of so-called clique trees (arranging the maximal cliques of the graph in a tree). The hypergraph-theoretical background of chordal graphs is given by α-acyclic hypergraphs which play an important role in the theory of relational database schemes. Various desirable properties of such schemes can be expressed in terms of various levels of acyclicity of hypergraphs [203, 204]: Chordal graphs correspond to α-acyclic hypergraphs, dually chordal graphs correspond to the dual hypergraphs of α-acyclic hypergraphs, strongly chordal graphs correspond to β-acyclic hypergraphs (which are equivalent to totally balanced hypergraphs), ptolemaic graphs correspond to γ-acyclic hypergraphs, and block graphs correspond to Berge-acyclic hypergraphs. Actually, tree structure of hypergraphs was captured as arboreal hypergraphs by Berge [32, 33]; a hypergraph is α-acyclic if and only its dual is arboreal. One of the chapters is devoted to structural properties of strongly chordal graphs and chordal bipartite graphs which are closely related to each other. Another chapter is dealing with distance-hereditary graphs and important subclasses. We discuss also another width parameter of graphs, namely clique-width, and its relationship to treewidth as well as its algorithmic applications. Very similar to treewidth, it is known that whenever a problem is expressible in a certain kind of Monadic Second Order Logic, and one deals with a class of graph whose clique-width is bounded by a constant then the problem is efficiently solvable on this class. This is one of the main 5

6 reasons for the great interest in treewidth and clique-width of (special) graphs. In general, it is NP-hard to determine the clique-width of a graph, and for many important graph classes, the clique-width is unbounded. For some interesting classes, however, clique-width is bounded. One of the chapters deals with the concept of leaf powers which is motivated by phylogeny and is based on distances of leaves in trees. Leaf powers form an important subclass of strongly chordal graphs. Finally, we discuss some other graph parameters, namely, tree-length and tree-breadth of a graph, tree-distortion and tree-stretch of a graph, and Gromov s hyperbolicity of a graph. All these parameters try to capture and measure tree likeness of a graph from a metric point of view. The smaller such a parameter is for a graph, the closer the graph is to a tree metrically. Graphs for which such parameters are bounded by small constants have many algorithmic advantages; they allow efficient approximate solutions for a number of optimization problems. Note also that recent empirical and theoretical work has suggested that many real-life complex networks and graphs arising in Internet applications, in biological and social sciences, in chemistry and physics have tree-like structures from a metric point of view. A first version of this book appeared as Chapter 29 in [64]. 6

7 2 Basic Notions and Results 2.1 Basic Graph Notions and Results Throughout this book, let G = (V,E) be a finite undirected graph without self-loops and multiple edges with vertex set V and edge set E, and let V = n, E = m. For a vertex v V, let N(v) = {u uv E} denote the (open) neighborhood of v in G, and let N[v] = {v} {u uv E} denote the closed neighborhood of v in G. Vertices x and y are adjacent in G if xy E, otherwise they are nonadjacent. We also say that adjacent vertices see each other, and nonadjacent vertices miss each other. Two vertices x,y V are true twins if N[x] = N[y], i.e., x and y have the same neighbors and are adjacent to each other. Two vertices are false twins if they have the same neighbors and are nonadjacent to each other. The degree deg G (v) of a vertex v is the number of its neighbors, deg G (v) = N(v). If N[v] = V, v is a universal vertex of G. A clique is a set of vertices which are mutually adjacent. An independent set (or stable set) in G is a set of vertices which are mutually nonadjacent. For graph G = (V,E), the complement graph G has the same vertex set V, and two distinct vertices are adjacent in G if and only if they are nonadjacent in G. For U V, let G[U] denote the subgraph of G induced by U. We write G U for G[V \U], and we write G u for G {u}. Throughout this book, all subgraphs are understood to be induced subgraphs. Let F denote a set of graphs. A graph G is F-free if none of its induced subgraphs is in F. A graph is connected (or 1-connected) if there is a path between every pair of distinct vertices. The maximal connected subgraphs are the connected components of G. U V isacutsetingifg U hasmoreconnected componentsthang. Ak-cutinaconnected graph is a cutset with k vertices; a 1-cut is also called a cut-vertex. G is k-connected if it has no cutset with at most k 1 vertices. For a positive integer k, a k-connected component in a graph G is a maximal (induced) k-connected subgraph of G; the 1- connected components of G are the usual connected components, and the 2-connected components of G are also called blocks of G. For graphs G 1 = (V 1,E 1 ), G 2 = (V 2,E 2 ), the graph G 1 G 2 (G 1 G 2, respectively) has vertex set V 1 V 2 (V 1 V 2, respectively) and edge set E 1 E 2 (E 1 E 2, respectively). Let d G (x,y) (or d(x,y) for short if G is understood) be the length, i.e., number of edges, of a shortest path in G between x and y. For k 1, let G k = (V,E k ) with xy E k if and only if d G (x,y) k denote the k-th power of G. If G = H k then H is a k-th root of G; if k = 2 then H is called a square root of G. For k 1, let P k denote a chordless path with k vertices and k 1 edges, and for k 4, let C k denote a chordless cycle with k vertices and k edges. A hole is a chordless cycle C k with k 5, and an odd hole is a chordless cycle C 2k+1 with k 2. An antihole is the complement graph of a hole, and an odd antihole is the complement graph of an odd hole. 7

8 Definition 2.1 Let G be a finite undirected graph. (i) G is chordal if for every k 4, G is C k -free. (ii) G is weakly chordal if for every k 5, G and G are C k -free. Clearly, every chordal graph is weakly chordal since C 5 is isomorphic to C 5 and C k contains C 4 for every k 6. A complete bipartite graph with r vertices in one color class and s vertices in the other color class is denoted by K r,s ; K 1,3 is also called claw. Let S k denote the (complete) sun with 2k vertices u 1,...,u k and w 1,...,w k such that u 1,...,u k is a clique, w 1,...,w k is a stable set and for all i {1,...,k}, N(w i ) = {u i,u i+1 } (index arithmetic modulo k). The diamond has four vertices and exactly one pair of nonadjacent vertices, i.e., it is a K 4 minus one edge, denoted K4. In general, for k 4, a clique with k vertices minus an edge is denoted K k ; it is the (k 2)-th power of the induced path P k. The gem (see Figure 20) has five vertices such that four of them induce a P 4 and the fifth is adjacent to all of them. Let O = {o 1,...,o n } be a set of objects for which the notion of intersection makes sense. A graph G = (V,E) with V = {v 1,...,v n } is the intersection graph of O if for i j, v i v j E if and only if o i o j. Examples of such classes are chordal graphs and interval graphs as we will see in section 3. A vertex set I is independent (or stable) in G if the vertices in I are pairwise nonadjacent. The independence number α(g) is the maximum cardinality of an independent vertex set in G. Correspondingly, a vertex set is a clique in G if it is an independent set in G. The clique number ω(g) is the maximum cardinality of a clique in G, i.e., ω(g) = α(g). An assignment of labels (called colors) to vertices of G = (V,E) is an admissible vertex coloring (vertex coloring for short) of G if no two vertices of G share the same label. It can be interpreted as a partition of V into independent vertex set in G. The chromatic number χ(g) is the minimum number of colors for any vertex coloring of G. Correspondingly, the clique cover number κ(g) of G is minimum number of cliques for any clique partition of G, i.e., κ(g) = χ(g). The coloring problem of a graph is how to assign a minimum number of colors to the vertices such that adjacent vertices get different colors. The chromatic number χ(g) of the graph G is the minimum number of colors needed to color G. Obviously, for every graph G, ω(g) χ(g) holds. A graph is called χ-perfect if for every induced subgraph G of G (including G itself), ω(g ) = χ(g ) holds. Let κ(g) = χ(g). Obviously, α(g) κ(g) holds. A graph is called κ-perfect if for every induced subgraph G of G (including G itself), α(g ) = κ(g ) holds. The following theorem is a celebrated result by Laszló Lovász (see e.g. [93]): 8

9 Theorem 2.1 (Perfect Graph Theorem) A graph is χ-perfect if and only if it is κ-perfect. Now, G is called perfect if G is χ-perfect (κ-perfect). Corollary 2.1 A graph G is perfect if and only if its complement graph G is perfect. The famous Strong Perfect Graph Theorem was shown by Chudnovsky, Robertson, Seymour and Thomas in [143]: Theorem 2.2 (Strong Perfect Graph Theorem) A graph is perfect if and only if it is odd-hole-free and odd-antihole-free. See e.g. [93] for many important subclasses of perfect graphs; one of them is the class of weakly chordal graphs [267]. Various graph parameters such as α(g), ω(g), χ(g), and κ(g) are hard to determine; the corresponding decision problems are NP-complete (see e.g. [224]). A matched co-bipartite graph is a co-bipartite graph, consisting of two disjoint cliques X and Y of the same size and a perfect matching between them. A vertex z V \{x,y} distinguishes two vertices x,y V if z is adjacent to exactly one of them, say zx E and zy / E. A vertex subset U V is a module in G if no vertex from V \ U distinguishes two vertices in U. A nontrivial module is a module with at least two but not all vertices. A nontrivial module of a graph is maximal if there is no other nontrivial module of the graph containing it. A clique module in G is a module which induces a clique in G. Distinct vertices x,y V are true twins in G if N[x] = N[y]. They are false twins if N(x) = N(y), i.e., they have the same neighbors and are nonadjacent to each other. Disjoint vertex sets X,Y form a join (cojoin, respectively), denoted by X 1 Y (X 0 Y, respectively), if for all pairs x X, y Y, xy E (xy E, respectively) holds Line graphs Let G = (V,E) be a graph. The line graph of G is the graph L(G) = (E,E ) with ee E if and only if e e and e e. Theorem 2.3 (Beineke [30]) Agraph is a line graphif and onlyif itdoes notcontain any induced subgraph isomorphic to one of the graphs in Figure 1. See e.g. [359] for other characterizations of line graphs. 9

10 Figure 1: The forbidden induced subgraphs for line graphs Transitively orientable graphs A graph G = (V,E) is called transitively orientable if its edge set E can be oriented as E in such a way that for all oriented edges (x,y), (y,z) E, (x,z) E holds Some Basic Hypergraph Notions and Properties A pair H = (V,E) is a (finite) hypergraph if V is a finite vertex set and E is a collection of subsets of V (the edges or hyperedges of H). Hypergraphs are a natural generalization of undirected graphs; unlike edges, hyperedges are not necessarily two-elementary. In many cases, hyperedges containing exactly one vertex (so-called loops) are excluded. Equivalently, a hypergraph H = (V,E) with V = {v 1,...,v n } and E = {e 1,...,e m } can be described by its n m vertex-hyperedge incidence matrix M(H) with entries m ij {0,1} and m ij = 1 v i e j for i {1,...,n} and j {1,...,m}. Subsequently, we collect some basic notions and properties - see e.g. [33]. Definition 2.2 A hypergraph H = (V,E) is simple if it has no repeated edges. Moreover, if no hyperedge e E is properly contained in another hyperedge e E then H is called a Sperner family or clutter. In the database community (see e.g. [28]), clutters are called reduced hypergraphs. Definition 2.3 Let H = (V,E) be a finite hypergraph. (i) The subhypergraph induced by the subset A V is the hypergraph H[A] = (A,E A ) with edge set E A = {e A e E}. 10

11 (ii) The partial hypergraph given by the edge subset E E is the hypergraph with the vertex set E and the edge set E. Note that both restrictions A V and E E can be combined in a subhypergraph H [A] = (A,E A ) with edge set E A = {e A e E E} called partial subhypergraph in [33]. The partial hypergraphs [33] are called subhypergraphs in [203]. Since this may cause confusion, we also use the name edge-subhypergraphs for partial hypergraphs and vertexsubhypergraphs in case (i). Dualization is a classical concept which is well known from geometry; there, points and hyperplanes exchange their role. Here, dualization means that vertices and hyperedges exchange their role: Definition 2.4 Let H = (V,E) be a finite hypergraph. For v V, let E v = {e E v e}. The dual hypergraph H = (E,E ) of H has vertex set E and hyperedge set E = {E v v V}. If the hypergraph H is given in terms of its incidence matrix M(H) then the incidence matrix of the dual of H is the transposal of M(H): M(H ) = (M(H)) T. Evidently, the dual of the dual of H is isomorphic to H itself since for the twofold transposal, ((M(H)) T ) T = M holds. Thus: Proposition 2.1 (H ) H. Graphs and hypergraphs are closely related to each other. The next definition represents two examples. Definition 2.5 Let H = (V,E) be a finite hypergraph. (i) The 2-section graph 2SEC(H) of H has the vertex set V, and two vertices u,v are adjacent if u and v are contained in a common hyperedge: e E such that u,v e. (ii) The line graph L(H) = (E,F) is the intersection graph of E, i.e., for any e,e E with e e, ee F e e. The 2-section graph of H is denoted by [H] 2 in [33]; the line graph is also called representative graph in [33]. Again, these notions have different names in different communities; the 2-section graph is also called adjacency graph in [246], primal graph [247] or Gaifman graph [222] and has no name but is denoted by G(H) in [28]. The line graph is also called dual graph in [176, 352]. The following isomorphism is easy to see: 11

12 Proposition 2.2 2SEC(H) L(H ). A subfamily E E is called pairwise intersecting if for all e,e E, e e. Definition 2.6 Let H = (V,E) be a hypergraph. (i) H is conformal if every clique C in 2SEC(H) is contained in a hyperedge e E. (ii) H has the Helly property if every pairwise intersecting subfamily E E has nonempty total intersection: E. The following is easy to see: Proposition 2.3 H is conformal if and only if H has the Helly property. Proof. = : Assume that H is conformal, i.e., every clique Q in in 2SEC(H) is contained inahyperedge e Q E. Weshow that H has thehelly property: Let V V be a family of vertices such that for all x,y V, E x E y. Then V is a clique in H, and since H is conformal, there is a hyperedge e V such that V e V. Now e V x V E x and thus, H has the Helly property. = : Assume that H = (E,{E v v V}) has the Helly property. We show that H is conformal: Let Q be a clique in 2SEC(H). Then for all x,y Q, there is a hyperedge e x,y E containing both of them, i.e., E x E y. Since H has the Helly property, we have x Q E x. Thus, there is a hyperedge e Q E with Q e Q. The next theorem gives a polynomial time criterion for testing the Helly property of a hypergraph. It is closely related to an earlier criterion for conformality given by Gilmore which will be mentioned in Theorem 2.5. For a hypergraph H = (V,E) and for any 3-elementary set A = {a 1,a 2,a 3 } V, let E A denote the set of all hyperedges e E such that e A 2. Theorem 2.4 (Ryser [406], Berge, Duchet; see [33]) A hypergraph H = (V, E) has the Helly property if and only if for all 3-elementary sets A = {a 1,a 2,a 3 } V, the total intersection of all hyperedges containing at least two vertices of A is nonempty: EA. Proof. = : Let H be a hypergraph with the Helly property, let A = {a 1,a 2,a 3 } V, and let E A be as above. Then for all e i,e j E A, e i e j and thus, E A since H has the Helly property. = : Now assume that {e 1,...,e l } E is a collection of pairwise intersecting hyperedges. If l = 2 then obviously their total intersection is nonempty; thus let l > 2. We assume inductively that the assertion of nonempty total intersection is true for less than l hyperedges with pairwise nonempty intersection. 12

13 Then by the induction hypothesis, e 1... e l 1 ; let a 1 e 1... e l 1. Moreover, e 2... e l ; let a 2 e 2... e l. Finally e 1 e l ; let a 3 e 1 e l. Let A := {a 1,a 2,a 3 }. It is easy to see that in the case A < 3 we are done. Thus let A = 3; every e i, i = 1,...,l, contains at least two elements from A, and by assumption, e 1... e l. An obvious consequence of Theorem 2.4 is: Corollary 2.2 Testing the Helly property for a given hypergraph can be done in polynomial time. Corollary 2.3 Every collection of subtrees of a tree has the Helly property. Proof. Let T be a tree with at least three vertices (otherwise the assertion is obviously fulfilled), and let a,b,c be any three vertices in T. We consider the set of all subtrees of T containing at least two of the vertices a,b,c. Let P(x,y) denote the uniquely determined path in the tree T between x and y. Let x 0 denote the last vertex in P(a,b) P(b,c) (this intersection contains at least vertex b). Then P(a,c) consists of P(a,x 0 ) followed by P(x 0,c). Thus the three paths P(a,b), P(b,c) and P(a,c) have vertex x 0 in common, i.e., x 0 is contained in every subtree of T which contains at least two of the vertices a,b,c. Thus, by Theorem 2.4, every system of subtrees has the Helly property. A nice inductive proof of Corollary 2.3 is given in a script by Alexander Schrijver: The induction is on V(T). If V(T) = 1 then the assertion is trivial. Now assume V(T) 2, and let S be a collection of pairwise intersecting subtrees of T. Let t be a leaf of T. If there exists a subtree of T consisting only of t, the assertion is trivial. Hence we may assume that each subtree in S containing t also contains the neighbor of t in T. So, after deleting t from T and from all subtrees in S, this collection is still pairwise intersecting, and the assertion follows by induction. Actually, Theorem 2.4 is formulated in a more general way in [33]; there are various interesting generalizations of the Helly property. According to Proposition 2.3, Theorem 2.4 can be dualized as follows: Theorem 2.5 (Gilmore, see [33]) Let H = (V,E) be a hypergraph. H is conformal if and only if for all 3-elementary edge sets A = {e 1,e 2,e 3 } E of hyperedges, there is a hyperedge e E with (e 1 e 2 ) (e 1 e 3 ) (e 2 e 3 ) e. Proof. = : Obviously, (e 1 e 2 ) (e 1 e 3 ) (e 2 e 3 ) is a clique in the 2-section graph 2SEC(H) of H. By conformality, there is a hyperedge e with (e 1 e 2 ) (e 1 e 3 ) (e 2 e 3 ) e. = : Let A = {e 1,e 2,e 3 } E and let E u be a hyperedge in H containing at least two of e 1,e 2,e 3. Then u (e 1 e 2 ) (e 1 e 3 ) (e 2 e 3 ) and thus also u e. Thus, e is in the total intersection of all hyperedges E u which contain at least two of e 1,e 2,e 3. 13

14 Then by Theorem 2.4, H has the Helly property and thus, by Proposition 2.3, H is conformal. There is a third type of graphs derived from a hypergraph H = (V,E), namely the bipartite vertex-edge incidence graph I(H) (which is a reformulation of the incidence matrix of H in terms of a bipartite graph). The two color classes of I(H) are the sets V and E, respectively, and a vertex v and an edge e are adjacent if and only if v e. More formally: Definition 2.7 Let H = (V,E) be a finite hypergraph. In the bipartite incidence graph I(H) = (V,E,I) of H, v V and e E are adjacent if and only if v e. In the other direction, namely from graphs to hypergraphs, the most basic constructions are the following: Definition 2.8 Let G = (V,E) be a graph. (i) The clique hypergraph C(G) consists of the -maximal cliques of G. (ii) The neighborhood hypergraph N(G) consists of the closed neighborhoods N[v] of all vertices v in G. (iii) The disk hypergraph D(G) consists of the iterated closed neighborhoods N i [v], i 1, of all vertices v in G, where N 1 [v] := N[v] and N i+1 [v] := N[N i [v]]. Note that in general, the neighborhood hypergraph N(G) is not simple since different vertices can have the same closed neighborhood in G. The following is easy to see: Proposition 2.4 N(G) is self-dual, i.e., (N(G)) N(G). Moreover, the 2-section graph of C(G) is isomorphic to G and thus, C(G) is conformal. Note that a hypergraph uniquely determines its 2-section graph but not vice versa. Lemma 2.1 Every conformal Sperner hypergraph H = (V, E) is the clique hypergraph of its 2-section graph 2SEC(H): H = C(2SEC(H)). Proof. Let H be conformal and Sperner. We show: 1) For every e E, e is a maximal clique in 2SEC(H): Obviously, eisacliquein2sec(h)andthus, thereisamaximalcliquec in2sec(h) with e C. Since H is conformal, there is an e E with C e, i.e., e C e and since H is Sperner, e = C = e follows. 2) For every maximal clique C in 2SEC(H), C E holds: By conformality of H, there is e E with C e, and since e is a clique in 2SEC(H), thereis amaximal clique C in2sec(h)withe C, i.e., C e C. Bymaximality of C, C = e = C follows. 14

15 Recall that the square G 2 = (V,E 2 ) of a graph G is defined as xy E 2 for x y if and only if d G (x,y) 2 (see Section 2). The following is easy to see: Proposition 2.5 G 2 L(N(G)). Definition 2.9 For graph G = (V,E), let B(G) = (V,V,F) denote the bipartite graph with two disjoint copies V and V of V, and for v V and w V, v w F if and only if either v = w or vw E. The following is easy to see: Proposition 2.6 B(G) I(N(G)). The line graph of C(G) is the classical clique graph operator in graph theory: Definition 2.10 Let G be a graph. (i) The clique graph K(G) of G is defined as K(G) = L(C(G)). (ii) G is a clique graph if there is a graph G such that G is the clique graph of G, i.e., G = K(G ). Theorem 2.6 (Roberts, Spencer [397]) A graph G is a clique graph if and only if some class of complete subgraphs of G covers all edges of G and has the Helly property. See [93, 359] and in particular the survey [417] by Szwarcfiter for more details on clique graphs. Recognizing whether a graph is a clique graph is NP-complete [5]. 15

16 3 Chordal Graphs, Subclasses and Variants In this section, we collect some notions and well-known facts on chordal graphs which are described in various monographs; see e.g. [236] and the survey [93] as well as [359] for details. In order to make this section self-contained, we briefly repeat some of the basic definitions and properties. Throughout this section, let G = (V,E) be a finite undirected graph which is simple (i.e., loop-free and without multiple edges). 3.1 Chordal Graphs Structural Properties of Chordal Graphs Recall Definition 2.1 for chordal graphs. Obviously, trees and forests are chordal since they are cycle-free for any cycle length. Chordal graphs have a nice separator property which was found by Dirac [183]. Definition 3.1 (i) The vertex set S V is a separator (or cutset) for nonadjacent vertices a,b V (a-b-separator) if a and b are in different connected components in G[V \S]. (ii) S is a minimal a-b-separator if S is an a-b-separator and no proper subset of S is an a-b-separator. (iii) S is a (minimal) separator if there are vertices a,b such that S is a (minimal) a-b-separator. Theorem 3.1 (Dirac [183]) A graph G is chordal if and only if every minimal separator in G induces a clique. Proof. = : Let (a,x,b,y 1,y 2,...,y k,a) be a simple cycle of G, k 1. Every cycle of length 4 is of this form (if there is any cycle in G). If ab E then the cycle contains a chord. If ab / E then the vertices a and b can be separated. Every minimal a-b-separator contains x and y i for some i, 1 i k, and since the separator induces a clique, xy i E, thus the cycle contains a chord. = : Let S be a minimal a-b-separator and let G[A] (G[B], respectively) the connected component of G[V \ S] containing a (b, respectively). Since S is a minimal separator, every x S is adjacent to a vertex in G[A] and to a vertex in G[B]. Thus, for each pair x,y S with x y, there are paths P 1 = (x,a 1,...,a r,y) and P 2 = (y,b 1,...,b s,x), a i G[A],b j G[B], i {1,...,r}, j {1,...,s}. Without loss of generality, choose such paths of minimal length. Then (x,a 1,...,a r,y,b 1,...,b s,x) is a simple cycle of length 4. By assumption, this cycle must have a chord. Since, moreover, the paths P 1,P 2 have minimal length, the only possible chord is the edge xy E. Thus, S induces a clique. 16

17 Definition 3.2 (i) A vertex v V is simplicial in G if N(v) induces a clique in G. (ii) An ordering (v 1,...,v n ) of the vertices of V is a perfect elimination ordering (p.e.o.) of G if for all i {1,...,n}, the vertex v i is simplicial in the remaining subgraph G i := G[{v i,...,v n }]. Obviously, the notion of a simplicial vertex generalizes leaves in trees. Lemma 3.1 (Dirac [183]) Every chordal graph with at least one vertex contains a simplicial vertex. If G is not a clique then G contains at least two nonadjacent simplicial vertices. Proof. Let G = (V,E) be a chordal graph. If G is a complete graph then the assertion is trivially fulfilled. Now let a,b V, ab / E, and, as induction hypothesis, let Lemma 3.1befulfilledforallgraphswithlessvertices thang. LetS beaminimal a-b-separator with a G[A], b G[B], where G[A] (G[B], respectively) is the connected component of G[V \S] containing a (containing b, respectively). By Theorem 3.1, S is a clique. By induction hypothesis, G[A S] either contains two nonadjacent simplicial vertices, one of them being in A since S induces a clique, or G[A S] itself is a clique and every vertex of A is simplicial in G[A S]. Since N(A) A S, every vertex in A which is simplicial in G[A S] is also simplicial in G. Analogously, G[B] contains a vertex which is simplicial in G (and nonadjacent to any vertex in G[A]). Corollary 3.1 (Dirac [183]; Fulkerson, Gross [221]) G is chordal if and only if G has a perfect elimination ordering. Moreover, every simplicial vertex of a chordal graph G can be the first vertex of a perfect elimination ordering of G. Proof. = : Let G be chordal. Then by Lemma 3.1, G contains a simplicial vertex v. Since G[V \{v}] is also chordal, the repeated deletion of simplicial vertices leads to a perfect elimination ordering of G. = : Let (v 1,...,v n ) be a perfect elimination ordering of G and let C be a simple cycle of G, where v is the leftmost vertex of C in the perfect elimination ordering. Since N(v) C 2, the neighbors of v in C have a chord. Thus, G is chordal. Let s 1 = (a 1,...,a k ) and s 2 = (b 1,...,b l ) be vectors of positive integers; such vectors are used as vertex labels in Algorithm 3.1. Then s 1 is lexicographically smaller than s 2 (s 1 < s 2 ) if either there is an index i min(k,l) such that a i < b i and a j = b j for all j, 1 j i 1 or k < l and a i = b i for all i with 1 i k. If s = (a 1,...,a k ) is a vector of positive integers and p is a positive integer then s+p := (a 1,...,a k,p). Let () denote the empty label. The subsequent algorithm Lexicographic Breadth-First Search (LexBFS) assigns a distinct number to each vertex of V depending on its label and, during the procedure, updates labels of neighbors as follows: 17

18 Algorithm 3.1 (Lexicographic Breadth-First Search (LexBFS)) Input: A graph G = (V,E) Output: A LexBFS ordering σ = (v 1,...,v n ) of V. begin for all v V do l(v) := (); for k := V = n downto 1 do choose a vertex v with lexicographically largest label l(v); set σ(k) := v; for all u V N(v) do l(u) := l(u)+k; set V := V \{v}; endfor; end Theorem 3.2 (Rose, Tarjan, Lueker [404]) For a chordal graph G, every LexBFS ordering of G is a p.e.o. of G. This leads to a linear time recognition algorithm for chordality of graphs [404] as follows: 1. For a given graph G = (V,E), determine a LexBFS ordering σ. 2. Check if σ is a p.e.o. of G; if yes, G is chordal, otherwise G is not chordal. This can be done in linear time (see e.g. [236] for details); LexBFS as partition refinement explains why step 1 can be done in linear time. Subsequently, for an ordering σ = (v 1,...,v n ) of V and two vertices x = v i and y = v j, x < y means that i < j. LexBFS orderings of graphs can be characterized by the following 4-point condition: For all a < b < c with ac E,bc / E there is d > c with bd E and ad / E. (1) Theorem 3.3 ([68]) For any graph G = (V,E), an ordering σ = (v 1,...,v n ) of V is a LexBFS ordering of G if and only if σ fulfills the 4-point condition (1). See [149] for a survey on LexBFS and other vertex orderings. There is another linear-time algorithm for recognizing chordal graphs called Maximum Cardinality Search (MCS) introduced in [421] (see also [236]): Algorithm 3.2 (Maximum Cardinality Search (MCS)) Input: A graph G = (V,E) Output: A MCS ordering σ = (v 1,...,v n ) of V. 18

19 begin for k := V = n downto 1 do choose a vertex v with a maximal number of numbered neighbors; set σ(k) := v; V := V \{v}; endfor; end For a collection T of subtrees of a tree T, let the vertex intersection graph G T of T be the graph having the elements of T as its vertices, and two subtrees t and t from T are adjacent in G T if they share a vertex in T. Proposition 3.1 The vertex intersection graph of a collection of subtrees in a tree is chordal. Proof. Let G = (V,E) be the vertex intersection graph of a collection of subtrees in a tree T. Suppose G contains a chordless cycle (v 0,v 1,...,v k 1,v 0 ) with k > 3 corresponding to the sequence of subtrees T 0,T 1,...,T k 1,T 0 of the tree T; that is, T i T j if and only if i and j differ by at most one modulo k. All index arithmetic will be done modulo k. Choose a point a i from T i T i+1 (i = 0,...,k 1). Let b i be the last common point on the (unique) simple paths from a i to a i 1 and a i to a i+1. These paths lie in T i and T i+1, respectively, so that b i also lies in T i and T i+1. Let P i+1 be the simple path connecting b i to b i+1 in T. Clearly P i T i, so P i P j = for i and j differing by more than 1 mod k. Moreover, P i P i+1 = {b i } for i = 0,...,k 1. Thus, i P i is a simple cycle in T, contradicting the definition of a tree. The tree structure of chordal graphs is described in terms of so-called clique trees of the maximal cliques of the graph; see Theorem 3.4. Definition 3.3 Let C(G) denote the family of -maximal cliques of graph G = (V,E). A clique tree T of G has the maximal cliques of G as its nodes, and for every vertex v of G, the maximal cliques containing v form a subtree of T. This notion will be generalized in Chapter 4 on hypergraphs; it can be taken for defining α-acyclicity of a hypergraph (see Definition 4.4). The existence of a clique tree characterizes chordal graphs: Theorem 3.4 (Buneman [113], Gavril [228], Walter [432]) A graph is chordal if and only if it has a clique tree. Proof. = : Assume that G has a clique tree T. If T has only one node then G is a clique and thus chordal. Now let T have k > 1 nodes and assume as induction hypothesis that the assertion is true for clique trees with less than k nodes. Let C be a leaf node in T, let C be its neighbor in T, let V C be the subset of G vertices occuring only in C, and let T be the clique tree restricted to V \V C. 19

20 V C must be nonempty since otherwise, C C which is impossible by maximality of the cliques. Now start a p.e.o. of G with the vertices of V C and then continue with a p.e.o. for G V C which must exist since T has less nodes than T. = : For this direction, we assume that G is chordal, and let σ = (v 1,...,v n ) be a p.e.o. of G. We inductively assume that G := G[{v 2,...,v n }] has a clique tree T. Since v 1 is simplicial in G, N(v 1 ) is a clique but not necessarily a maximal one in G. In the case that it is a maximal clique in G represented by the node Q in T then add v 1 to Q, and now T is a clique tree of G. In the other case when N(v 1 ) is not a maximal clique in G, it is contained in a maximal clique of G represented by node Q in T. Now add a new node N[v 1 ] to T and connect it to Q; obviously, this extension of T is a clique tree of G. The following version described by Spinrad in [414] leads to linear time construction of clique trees: We construct a clique tree for the subgraph G i = G[v i,...,v n ] for all vertices, starting with i = n and ending with i = 1. Let C i be the clique consisting of v i and all neighbors v j of v i, j > i. After each vertex v i is processed, v i is given a pointer to the clique C i in the tree. We note that vertices may be added to this clique later in the algorithm, but v i will always point to a clique which contains C i. Let v i be the next vertex considered, and assume we know the clique tree on the graph induced by vertices v i+1,...,v n. We need to add C i to the clique tree. Let v j be the first (i.e., leftmost) vertex of C i on the right of v i in σ. If C i = C j + 1, and the clique pointed to by v j is equal to C j then we add v i to this clique; in other words, C i replaces C j in the tree. Otherwise, add C i as a new node of the tree. Connect C i to the tree by adding an edge from C i to the clique pointed to by v j. To see that the algorithm is correct, it is sufficient to look at two cases. Either C j is a maximal clique in G i+1 = G[{v i+1,...,v n }] or it is not. If C j is a maximal clique, it clearly must be replaced by C i if C j is contained in C i, which occurs if C i = C j {v i }, and the algorithm does this correctly. If C j is not a maximal clique in G i+1 or C i does not contain C j, then C i cannot contain any maximal clique of G i+1, and must be added as a new node. All elements of C i v i are in the clique pointed to by v j, so the subtrees generated by the occurrences of all vertices remain connected. Since a p.e.o. of a chordal graphcanbe determined inlinear time, the proofof Theorem 3.4 implies: Theorem 3.5 Given a chordal graph G = (V,E), a clique tree of G can be constructed in linear time O( V + E ). A consequence of Theorem 3.4 and Proposition 3.1 is: Corollary 3.2 (Buneman [113], Gavril [228], Walter [432]) A graph is chordal if and only if it is the intersection graph of certain subtrees of a tree. Proof. = : If G is chordal then by Theorem 3.4, it has a clique tree, say T. Thus, for an edge xy in G, the subtree T x of maximal cliques containing x, and the corresponding subtree T y intersect in a maximal clique Q containing x and y. 20

21 = : If G is the intersection graph of certain subtrees of a tree then, by Proposition 3.1, G is chordal. Interestingly, a clique tree of a chordal graph G gives also the minimal separators of G: Lemma 3.2 ([26, 274]) Let G = (V G,E G ) be a chordal graph with clique tree T = (C(G),E T ). Then S V G is a minimal separator in G if and only if there are maximal cliques Q i,q j of G with Q i Q j E T such that S = Q i Q j Algorithmic Problems for Chordal Graphs The specific structure of chordal graphs and in particular the perfect elimination orderings allow to solve various problems efficiently which is well described in [236] for the basic problems Maximum Clique, Maximum Independent Set, Chromatic Number and Clique Covering. A good example is the use of p.e.o. for determining the chromatic number χ(g) and the clique number ω(g) of a given chordal graph G; obviously, for every graph, ω(g) χ(g). Let G = (V,E) be a chordal graph with perfect elimination ordering σ = (v 1,...,v n ) of G. The following algorithm using σ from right to left colors the vertices of G with ω(g) colors. Starting with v n, a color is assigned to every v i, i = n,n 1,...,2,1, as follows: Since v i is simplicial in G i = G[{v i,...,v n )}], it has at most ω(g) 1neighbors in G i (which, as a clique, need N i (v i ) ω(g) 1 colors). Thus, at least one of the ω(g) colors is still available for v i ; take such a color for v i. This also shows: Corollary 3.3 Chordal graphs are perfect. Actually, this also follows from the fact that weakly chordal graphs are perfect [267] and chordal graphs are a subclass of them. While chordal graphs are a basic example for graphs with tree structure, weakly chordal graphs and perfect graphs do not seem to have any important tree structure. Another example for the use of perfect elimination orderings of chordal graphs is a linear time algorithm for Maximum Weight Independent Set on chordal graphs given by András Frank [219]. Let G = (V,E) be a chordal graph with perfect elimination ordering σ = (v 1,...,v n ) of G, and let S denote a maximum cardinality independent set in G. For the unweighted case, it is clear that without loss of generality, v 1 S since S N[v 1 ] = 1 (otherwise S would not be maximal) and if S N(v 1 ) = {x} then x can be replaced by v 1, i.e., S := (S \ {x}) {v 1 } is also a maximum cardinality independent set in G. Thus the algorithm for Maximum Cardinality Independent Set on chordal graphs goes along the p.e.o. σ from left to right, takes repeatedly a simplicial vertex v i and deletes its neighbors in G i = G[{v i,...,v n }]. For the weighted case the algorithm is a bit more complicated: Let G = (V,E) be a chordal graph with perfect elimination ordering (v 1,...,v n ) of G and w : V R + a nonnegative weight function on V. The algorithm of Frank efficiently constructs a maximum weight stable set I of G in the following way: 21

22 (0) I := ; all vertices in V are unmarked. (1) for i := 1 to n do if w(v i ) > 0 then mark v i and let w(u) := max(w(u) w(v i ),0) for all vertices u N i (v i ). (2) for i := n downto 1 do if v i is marked then let I := I {v i } and unmark all vertices u N(v i ). Theorem 3.6 ([219]) The algorithm described above is correct and runs in linear time. Proof. It is clear that the algorithm runs in linear time. For the correctness, we need the following (inductive) argument: As in the algorithm, let (v 1,...,v n ) be a p.e.o. of G and w a weight function on V. Now let w be the weight function resulting from step (1) of the algorithm for the neighbors of the simplicial vertex v 1. We claim: Claim α w (G) = w(v 1 )+α w (G v 1 ). This holds true by the following argument: Let S be a maximum-weight stable set in G and let C = N(v 1 ) denote the neighborhood of the simplicial vertex v 1 ; C is a clique. Clearly, S C 1. If for x C, w(v 1 ) > w(x) holds then clearly x / S - otherwise w(s) could be enlarged by replacing x by v 1. Thus, in this case, line (1) of the algorithm leads to w (x) = 0. If w(v 1 ) = w(x) then if x S, x can also be replaced by v 1. Case 1. v 1 S: Then N(v 1 ) S = and Claim holds since α w (G) = w(v 1 )+α w (G[V \N[v 1 ]]) = w(v 1 )+α w (G[V \N[v 1 ]]) = w(v 1 )+α w (G v 1 ) if S N(v 1 ) =. Case 2. v 1 / S: Then exactly one of the neighbors of v 1, say x, is in S (otherwise S would not be a maximal stable set), and now w (x) = w(x) w(v 1 ) > 0 holds and thus α w (G) = w(x)+α w (G[V \N[x]]) = w(v 1 )+(w(x) w(v 1 ))+α w (G[V \N[x]]) = w(v 1 )+w (x)+ α w (G[V \N[x]]) = w(v 1 )+α w (G v 1 ). Then the marking part of the algorithm delivers a maximum weight independent set in G. It should be mentioned that various important NP-complete problems remain NPcomplete for (subclasses of) chordal graphs; examples are the Hamilton Cycle problem [370] (which remains NP-complete for strongly chordal split graphs) and the Efficient Domination problem [412]. The Minimum Dominating Set problem remains NP-complete for split graphs [44]. 22

23 3.1.3 Convexity in Chordal Graphs and Powers of Chordal Graphs For graph G = (V,E), the k-th power G k of G is defined as G k = (V,E k ) with xy E k if x y and d G (x,y) k. An induced subgraph H of G = (V,E) is an isometric subgraph of G if for all pairs of vertices x,y V(H), their distance in H is the same as in G, i.e., d H (x,y) = d G (x,y). A vertex set S V is monophonically convex (m-convex for short) if for all pairs x,y S and every induced path P xy connecting x and y, each vertex of P xy is also contained in S. Lemma 3.3 (Farber, Jamison [208]) If G is a chordal graph with perfect elimination ordering (v 1,...,v n ) of G then for all i, 1 i n, {v i,...,v n } is m-convex and G i = G[{v i,...,v n }] is isometric in G. Chordal graphs are characterized by m-convexity in the following way: Theorem 3.7 (Farber, Jamison [208]) The following are equivalent: (i) G is chordal; (ii) for all v V the disk D 1 (v) = N[v] is m-convex; (iii) for all v V and all j > 0 the disk D j (v) with radius j is m-convex; (iv) for all K V which are m-convex N[K] is m-convex; (v) for all K V with K m-convex and all j > 0 the disks D j (K) are m-convex. While even powers of chordal graphs are in general not chordal (as the example of the square of the 4-sun shows), odd powers of chordal graphs are known to be chordal. An even stronger result is the following: Theorem 3.8 (Duchet[198]) If G k is chordal then G k+2 is chordal. LexBFS is helpful also in this case: Theorem 3.9 ([68]) For a chordal graph G, every LexBFS ordering of G is a p.e.o. of each odd power G 2k+1, k 1. The proof of Theorem 3.9 in [68] is based on m-convexity properties of chordal graphs as described above. Corollary 3.4 ([68]) A graph G is chordal if and only if every LexBFS ordering of G is a common p.e.o. of all odd powers of G. 23

24 This also implies that odd powers of a chordal graph are chordal. In [58], it is shown: Theorem 3.10 ([58]) There is a common p.e.o. for every chordal power of a chordal graph. Such a common p.e.o. is found by an algorithm called LMCS in [58] (which is a generalization of the Maximum Cardinality Search algorithm by Tarjan and Yannakakis [421]); the time bound, however, is not linear but O(nm). 3.2 Split Graphs Another subclass of chordal graphs which plays an important role in various contexts is the following: Definition 3.4 A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. Such a partition is called a split partition. It is easy to see that the complement of a split graph is a split graph as well, and split graphs are chordal. Theorem 3.11 (Főldes, Hammer [214]) The following conditions are equivalent: (i) G is a split graph. (ii) G and G are chordal. (iii) G is (2K 2,C 4,C 5 )-free. Proof. (ii) (iii) : If G and G are chordal then obviously G contains no induced 2K 2, C 4 and C 5. In the other direction, note that for every k 6, C k contains a 2K 2, and C 5 = C 5. Thus, if G contains no induced 2K 2, C 4 and C 5 then G and G are chordal. (i) = (ii) : If the vertex set V of G has a partition into a clique Q and a stable set S then obviously, every vertex in S is simplicial in G. Thus, a p.e.o. of G can start with all vertices of S and finish with all vertices of Q. Similar arguments hold for G, and thus G and G are chordal. (ii) = (i) : Suppose that G and G are chordal (or, equivalently, G contains no induced 2K 2, C 4 and C 5 ). If there is a vertex v V which is simplicial in G and G then N[v] is a clique and N[v] is a stable set giving the desired split partition. If there is a vertex v V which is neither simplicial in G nor simplicial in G then let a,b N(v) be vertices with ab / E and let c,d N[v] with cd E. Since G is 24