All the algorithms mentioned so far, except one, use variations on the ideas of cleaning and shortest paths detectors (or decomposition theorems), and that one exception is the algorithm for testing for T . There the approach is different. In order to be able to test for T , a slightly more general problem is studied: given a graph G, and three vertices v1, v2, v3 of G, does there exist an induced subgraph T of G, such that T is a tree and v1, v2, v3∈ V (T )? This is the three-in-a-tree problem.
It turns out that the answer to this question is “no” if and only if the graph admits a certain structure. This fact is then used to design a polynomial time algorithm for the three-in-a-tree problem. Now, if {v1, v2, v3} is a stable set of size three with a common neighbor w in G, the degree of each of v1, v2, v3in G \ {w} is one, and the degree of w in G is three, then the answer to the three-in-a-tree problem with input (G\{w}, v1, v2, v3) is “yes” if and only if G contains a theta using v1, v2, v3, w.
On the other hand, if {v1, v2, v3} is a clique of size three, and no vertex of G has two neighbors in it, then the answer to the three-in-a-tree problem with input (G, v1, v2, v3) is “yes” if and only if G contains a pyramid with base {v1, v2, v3}. Thus, the algorithm to solve the three-in-a-tree problem can be used, after some pre-processing, to test both for P and for T (and this is the only algorithm known to test for T ). This result is particularly pleasing from the point of view of a structural graph theorist, because this is one of the few times that a structure (and not just a decomposition) theorem and an algorithm appear together in the study of graphs with forbidden induced subgraphs.
As we have seen, the complexity of testing for F varies with F : for some families polynomial-time algorithms are known, while for others the problem can be shown to be NP-complete. An interesting open question is: what causes this difference? Can one characterize the families for which testing can be done efficiently?
2.5.6 Erd˝ os–Hajnal Conjecture and χ-Boundedness
As the results surveyed in Subsection 2.5.5 illustrate, structure theorems for F -free graphs tend to be complicated to state, difficult to prove, and hard to use. At the moment, we are nowhere near having a structural conjecture for excluding a general induced subgraph. But what if we lower our sights, and ask whether excluding a general induced subgraph guarantees that the graph has certain special properties that a general graph does not possess? In 1989, Erd¨os and Hajnal made a beautiful conjecture of this kind; it is now known as the Erd˝os–Hajnal Conjecture:
CONJECTURE
C1: [ErHa89] For every graph H, there exists a constant δ(H) > 0, such that every H-free graph G has either a clique or a stable set of size at least |V (G)|δ(H).
FACTS
In the same paper a partial result in this direction is proved, showing that for every H, H-free graphs behave differently from general graphs. It is a well-known theorem of Erd˝os that
F26: [Er47] There exist graphs on n vertices, with no clique or stable set of size larger than O(log n).
However,
F27: [ErHa89] For every graph H, there exists a constant c(H) > 0, such that every H-free graph G has either a clique or a stable set of size at least ec(H)
√
log|V (G)|.
DEFINITION
D50: Let us say that a graph H has the Erd˝os–Hajnal property if there exists a constant δ(H) > 0, such that every H-free graph G has either a clique or a stable set of size at least |V (G)|δ(H).
FACTS
F28: Clearly, H has the Erd˝os–Hajnal property if and only if Hc does.
Very few graphs have been shown to have the Erd˝os-Hajnal property.
F29: It is not difficult to show that all graphs on at most four vertices have the Erd˝os-Hajnal property.
A much more complicated argument is needed to show that:
F30: [ChSa08] The bull has the Erd˝os–Hajnal property.
In [AlPaSo01] it was shown that:
F31: [AlPaSo01] If H1, H2have the property, then so does every graph obtained from H1 and H2 by substitution.
Thus in order to prove Conjecture C1, it is enough to show that every prime graph has the Erd˝os–Hajnal property. However, this question is still open for C5, and for the five-vertex path. No prime graphs on at least six vertices have been shown to have the Erd˝os–Hajnal property.
Section 2.5. Structural Graph Theory 141
Tournaments
Next we introduce another version of Conjecture C1.
DEFINITION
D51: A tournament is a directed graph G where for every distinct u, v ∈ V (G), exactly one of the (ordered) pairs uv and vu belongs to E(G). If uv ∈ E(G), we say that u is adjacent to v.
D52: A tournament is transitive if it has no directed cycles (or, equivalently, no directed cycles of length three).
D53: For a tournament T , we denote by α(T ) the largest number of vertices in a transitive subtournament of T .
D54: For tournaments S and T , we say that T is S-free if no subtournament of T is isomorphic to S.
CONJECTURE
C2: [AlPaSo01] For every tournament S, there exists a constant δ(S) > 0, such that every S-free tournament T satisfies α(T ) ≥ |V (T )|δ(H).
FACT
It is also shown that:
F32: [AlPaSo01] Conjectures C1 and C2 are equivalent.
DEFINITION
D55: As with graphs, let us say that a tournament S has the Erd˝os–Hajnal prop-erty if there exists δ(S) > 0, such that every S-free tournament T satisfies α(T ) ≥
|V (T )|δ(H).
D56: Similarly to graphs, a tournament T is prime if there is no X ⊆ V (T ) with 1 < |X| < |V (T )| such that for every v ∈ V (T ) \ X, either v is adjacent to every vertex of X, or v is adjacent from every vertex of X.
For some reason, Conjecture C2 seems to be a little more approachable than Con-jecture C1:
FACT
F33: [BeChCh14] Unlike in the case of graphs, there is a known infinite family of prime tournaments, all of which have the Erd˝os–Hajnal property.
We refer the reader to [Ch13] for more information about recent progress on Con-jecture C1 and ConCon-jecture C2.
χ-Boundedness
Let us now consider another notion, related to Conjecture C1.
DEFINITIONS
D57: A class of graphs G is hereditary if H ∈ G for every G ∈ G and every induced subgraph H of G.
D58: We say that a hereditary graph G is χ-bounded if there exists a function f : N → N such that χ(G) ≤ f (ω(G)) for every G ∈ G. In this situation we call f a χ-bounding function for G.
EXAMPLES
E12: The class of F -free graphs is hereditary for every family F . E13: The class of perfect graphs is χ-bounded by the identity function.
E14: Suppose that for some graph H the class of H-free graphs has a χ-bounding function that is a polynomial. Then there exists t ≥ 1 such that χ(G) ≤ ω(G)tfor every H-free graph G. Since in every coloring of G, each color class has size at most α(G), it follows that
ω(G)tα(G) ≥ |V (G)|
and so G has either a clique or a stable set of size at least |V (G)|1/(t+1), and H has the Erd˝os–Hajnal property.
FACT
It is tempting to conjecture that the class of H-free graphs is χ-bounded for every H. However, this is false, as shown by the following theorem of Erd˝os [Er59]:
F34: [Er59] For every pair of integers k, g > 0 there exists a graph G with χ(G) > k and no cycle of length less than g.
Thus in order for the class of H-free graphs to be χ-bounded, H must contain no cycles (otherwise, every graph G with no cycle of length at most |V (H)| is H-free, and has ω(G) ≤ 2; and by Fact F34 there exist such graphs with arbitrarily large chromatic numbers). A famous conjecture of Gy´arf´as and Sumner [Gy75, Su81] states that this necessary condition is in fact sufficient:
CONJECTURE
C3: [Gy75, Su81] For every forest F , the class of F -free graphs is χ-bounded.
This conjecture is still open. Gy´arf´as [Gy75] proved that it holds when F is a path.
Kierstead and Penrice [KiPe90, KiPe94] and Scott [Sc97] made further progress. Some of the theorems in Subsection 2.5.2 and Subsection 2.5.3 also imply χ-boundedness results for certain classes of F -free graphs. We list some of them here.
Section 2.5. Structural Graph Theory 143
FACTS
F35: [AdChHaReSe08] For every even-hole-free graph G, χ(G) ≤ 2ω(G) − 1. This follows from Fact F8. Incidentally, the question of whether the class of odd-hole-free graphs is χ-bounded is still open.
F36: [ChSe10] If G is an induced subgraph of a connected claw-free graph G0 with α(G0) ≥ 3, then χ(G) ≤ 2ω(G). This follows from the main result of [ChSe08].
F37: [ChFr07] For every quasi-line graph G, χ(G) ≤ 32ω(G). This follows from the main result of [ChSe12].
For more results of this type, see [Vu13].
2.5.7 Well-Quasi-Ordering and Rao’s Conjecture
DEFINITION
D59: A quasi-order Q consists of a class E(Q) and a transitive reflexive relation which we denote by ≤ or ≤Q; and it is a well-quasi-order or wqo if for every infinite sequence qi(i = 1, 2 . . .) of elements of E(Q) there exist j > i ≥ 1 such that qi≤Qqj.
FACTS
One of the consequences of the Robertson–Seymour graph minor project is that:
F38: [RoSe04] The class of all graphs forms a well-quasi-order under minor contain-ment.
F39: The same is not true for induced subgraphs: the sequence C3, C4, ... is an infinite sequence of graphs, none of which is an induced subgraph of another.
This is disappointing, but S.B. Rao proposed the following “fix”:
DEFINITIONS
D60: Let us say two graphs G, G0 are degree-equivalent if they have the same vertex set, and for every vertex, its degrees in G and in G0 are equal.
D61: A graph H is Rao-contained in a graph G if H is isomorphic to an induced subgraph of some graph that is degree-equivalent to G.
FACT
In the early 1980s Rao [Ra81] conjectured the following , which was proved in [ChSe13]:
F40: [ChSe13] In any infinite set of graphs, there exist two, say G and H, such that H is Rao-contained in G.