The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs

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Abstract

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs.

Introduction

In this article, all graphs are finite and simple.

A prism is a graph made of three node-disjoint chordless paths P1=a1b1, P2=a2b2, P3=a3b3 of length at least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of the two triangles. Such a prism is also referred to as a 3PC(a1a2a3,b1b2b3) or a 3PC(Δ,Δ) (3PC stands for 3-path-configuration).

A pyramid is a graph made of three chordless paths P1=ab1, P2=ab2, P3=ab3 of length at least 1, two of which have length at least 2, node-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to a. Such a pyramid is also referred to as a 3PC(b1b2b3,a) or a 3PC(Δ,).

A theta is a graph made of three internally node-disjoint chordless paths P1=ab, P2=ab, P3=ab of length at least 2 and such that no edges exist between the paths except the three edges incident to a and the three edges incident to b. Such a theta is also referred to as a 3PC(a,b) or a 3PC(,).

A hole in a graph is a chordless cycle of length at least 4. Observe that the lengths of the paths in the three definitions above are designed so that the union of any two of the paths induce a hole. A wheel W=(H,c) is a graph formed by a hole H (called the rim) together with a node c (called the center) that has at least three neighbors in the hole.

A 3-path-configuration is a graph isomorphic to a prism, a pyramid or a theta. A Truemper configuration is a graph isomorphic to a prism, a pyramid, a theta or a wheel. They appear in a theorem of Truemper [36] that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities (3-path-configurations seem to have first appeared in a paper Watkins and Mesner [38]). (See Fig. 1.)

If G and H are graphs, we say that G contains H when H is isomorphic to an induced subgraph of G. We say that G is H-free if it does not contain H. We extend this to classes of graphs with the obvious meaning (for instance, a graph is (theta, wheel)-free if it does not contain a theta and does not contain a wheel).

Truemper configurations play an important role in the analysis of several important hereditary graph classes, as explained in a survey of Vušković [37]. Let us simply mention here that many decomposition theorems for classes of graphs are proved by studying how some Truemper configuration contained in the graph attaches to the rest of the graph, and often, the study relies on the fact that some other Truemper configurations are excluded from the class. The most famous example is perhaps the class of perfect graphs. In these graphs, pyramids are excluded, and how a prism contained in a perfect graphs attaches to the rest of the graph is important in the decomposition theorem for perfect graphs, whose corollary is the celebrated Strong Perfect Graph Theorem due to Chudnovksy, Robertson, Seymour and Thomas [10]. See also [34] for a survey on perfect graphs, where a section is specifically devoted to Truemper configurations. But many other examples exist, such as the seminal class of chordal graphs [17] (containing no holes and therefore no Truemper configurations), universally signable graphs [13] (which is exactly the class of graphs containing no Truemper configurations), even-hole-free graphs [15], [19] (containing pyramids but not containing thetas and prisms), cap-free graphs [14] (not containing prisms and pyramids, but containing thetas), ISK4-free graphs [21] (containing prisms and thetas but not containing pyramids), chordless graphs [22] (containing no prisms, pyramids and wheels, but containing thetas), (theta, triangle)-free graphs [29] (containing no prisms, pyramids and thetas), claw-free graphs [11] (containing prisms, but not containing pyramids and thetas) and bull-free graphs [7] (containing thetas and the prism on six nodes, but not containing pyramids and prisms on at least 7 nodes). In most of these classes, some wheels are allowed and some are not. In some of them (notably perfect graphs and even-hole-free graphs), the structure of a graph containing a wheel is an important step in the study of the class. Let us mention that the classical algorithm LexBFS produces an interesting ordering of the nodes in many classes of graphs where some well-chosen Truemper configurations are excluded [1]. Let us also mention that many subclasses of wheel-free graphs are well studied, namely unichord-free graphs [35], graphs that do not contain K4 or a subdivision of a wheel as an induced subgraph [21], graphs that do not contain K4 or a wheel as a subgraph [33], [3], propeller-free graphs [4], graphs with no wheel or antiwheel [23] and planar wheel-free graphs [2].

All these examples suggest that a systematic study of classes of graphs defined by excluding Truemper configurations is of interest. It might shed a new light on all the classes mentioned above and be interesting in its own right. In this paper we study two of such classes. Since there are four types of Truemper configurations, there are potentially 24=16 classes of graphs defined by excluding them (such as prism-free graphs, (theta, wheel)-free graphs, and so on). In one of them, none of the Truemper configurations are excluded, so it is the class of all graphs. We are left with 15 non-trivial classes where at least one type of Truemper configuration is excluded. One case is when all Truemper configurations are excluded. This class is known as the class of universally signable graphs [13] and it is well studied: its structure is fully described, and many difficult problems such as graph coloring, and the maximum clique and stable set problems can be solved in polynomial time for this class (see [1] for the most recent algorithms for them). So we are left with 14 classes of graphs, and to the best of our knowledge, they were not studied so far, except for one aspect: the complexity of the recognition problem is known for 11 of them. Let us survey this.

It is convenient to sum up in a table all the 16 classes. In Table 1, each line of the table represents a class of graphs defined by excluding some Truemper configurations. The first four columns indicate which Truemper configurations are excluded and which are allowed. The last columns indicates the complexity of the recognition algorithm and a reference to the paper where this complexity is proved. Lines with a reference to a theorem indicate a result proved here. For instance line 5 of the table should be read as follows: the complexity of deciding whether a graph is in the class of (theta, prism)-free graphs is O(n35) (throughout the paper, n stands for the number of nodes, and m for the number of edges of the input graph). Observe that a recognition algorithm for (theta, prism)-free graphs is equivalent to an algorithm to decide whether a graph contains a theta or a prism. Note that all the proofs of NP-completeness rely on a variant of a classical construction of Bienstock [5].

As already stated, 13 of the recognition problems of Table 1 are solved in previous work. In this paper and its subsequent part [26] we resolve the complexity of recognition of the remaining three classes. In this paper we give a polynomial time recognition algorithm for the following two classes: (theta, wheel, pyramid)-free and (theta, wheel, prism)-free graphs. In the first class, the only allowed Truemper configurations are prisms, and in the second, the only ones are pyramids. We therefore use the names only-prism and only-pyramid for these two classes. For the last problem from Table 1, namely the recognition of (theta, wheel)-free graphs, a similar approach is successful while being more complicated. This class is studied in a subsequent paper by the last three authors [26].

For each class, our recognition algorithm relies on a decomposition theorem for the class. In each case, this theorem fully describes the structure of the most general graph in the class, and could therefore be used to provide algorithms for several combinatorial optimization problems. This is done in Parts III and IV of this series (see [27] and [28]), where polynomial-time algorithms for finding maximum weighted clique and stable set, for optimal coloring and for the induced version of k-linkage problem (for k fixed) are obtained for the class of (theta, wheel)-free graphs. We note that among the 16 classes described in Table 1, only universally signable graphs (line 0 from the table) have a (previously known) decomposition theorem. All the other (previously known) polynomial time algorithms mentioned in Table 1 are based on a direct algorithm to detect the obstruction.

In Section 2, we give some notation and we describe the results, in particular we state precisely the decomposition theorems proved in the rest of the paper. In Section 3, we prove several lemmas needed in many places. In Section 4, we prove the decomposition theorem for only-prism graphs. In Section 5, we prove the decomposition theorem for only-pyramid graphs (note that the proof relies mostly on theorems proved previously in [19]). In Section 6, we prove that the 2-joins (a decomposition defined in the next section) that actually occur in our classes of graph have a special structure. In Section 7, we describe the recognition algorithms and show how the decomposition theorems that we prove can be transformed into structure theorems.

Section snippets

Main results

A path P is a sequence of distinct nodes p1p2pk, k1, such that pipi+1 is an edge for all 1i<k. Edges pipi+1, for 1i<k, are called the edges of P. Nodes p1 and pk are the ends of P. A cycle C is a sequence of nodes p1p2pkp1, k3, such that p1pk is a path and p1pk is an edge. Edges pipi+1, for 1i<k, and edge p1pk are called the edges of C. Let Q be a path or a cycle. The node set of Q is denoted by V(Q). The length of Q is the number of its edges. An edge e=uv is a chord of Q if u,vV(Q),

Preliminary lemmas

Lemma 3.1

If G is a diamond-free graph then every edge of G is contained in a unique maximal clique of G.

Proof

An edge uv is obviously in at least one maximal clique. If it is not unique, then let K and K be two distinct maximal cliques containing uv. Since by maximality KK, there exists wKK. By the maximality of K, there exists in K a non-neighbor w of w. So, {u,v,w,w} induces a diamond, a contradiction. □

When C and H are two disjoint sets of nodes of a graph (or induced subgraphs), we say that C is

Only-prism graphs

In this section we prove Theorem 2.5.

Lemma 4.1

Let G be an only-prism graph. Suppose that G contains two chordless paths P=xPyP and Q=xQzQ, of length at least 1, node disjoint, with no edges between them. Suppose x,y,zV(P)V(Q) are pairwise adjacent and such that N(x)(V(P)V(Q))={xP,xQ}, N(y)(V(P)V(Q))={yP} and N(z)(V(P)V(Q))={zQ}. Then, G has a clique cutset.

Proof

By Lemma 3.2, we may assume that G is diamond-free. So, by Lemma 3.1, there exists a unique maximal clique K of G that contains x, y and z.

Only-pyramid graphs

In this section we prove Theorem 2.7. The proof mostly relies on previously proved theorems and some terminology is needed to state them.

We say that a clique is big if it is of size at least 3. Let L be the line graph of a tree. By Theorem 2.3 and Lemma 3.1, every edge of L belongs to exactly one maximal clique, and every node of L belongs to at most two maximal cliques. The nodes of L that belong to exactly one maximal clique are called leaf nodes. In the graph obtained from L by removing all

2-Joins

In this section, we describe more closely the structure of the 2-joins and the almost 2-joins that actually occur in our classes of graphs. An almost 2-join with a split (X1,X2,A1,A2,B1,B2) in a graph G is consistent if the following statements hold for i=1,2:

  • (i)

    Every component of G[Xi] meets both Ai, Bi.

  • (ii)

    Every node of Ai has a non-neighbor in Bi.

  • (iii)

    Every node of Bi has a non-neighbor in Ai.

  • (iv)

    Either both A1, A2 are cliques, or one of A1 or A2 is a single node, and the other one is a disjoint union of

Algorithms

We are now ready to describe our recognition algorithms based on decomposition by clique cutsets and 2-joins. When a graph G has a clique cutset K, its node set can be partitioned into nonempty sets A, K, and B in such a way that there are no edges between A and B. We call such a triple a split for the clique cutset. When (A,K,B) is a split for a clique cutset of a graph G, the blocks of decomposition of G with respect to (A,K,B) are the graphs GA=G[AK] and GB=G[KB].

Lemma 7.1

Let G be a graph and (A,K,B

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    1

    Partially supported by Serbian Ministry of Education, Science and Technological Development project 174033.

    2

    Partially supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d'Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Also Université Lyon 1, université de Lyon.

    3

    Partially supported by EPSRC grant EP/K016423/1, and Serbian Ministry of Education and Science projects 174033 and III44006.

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