By we denote a path on vertices. A graph is -free if it does not contain an induced subgraph isomorphic to .
We are interested in classifying the complexity of fundamental computational problems, such asMaximum Weight Independent Set (MWIS), -Coloring for fixed or arbitrary , or Feedback Vertex Set, on various hereditary graph classes, in particular on -free graphs for small fixed graphs . As noted by Alekseev , MWIS is NP-hard on -free graphs unless every connected component of is a tree with at most three leaves. Similarly, -Coloring is known to be NP-hard on -free graphs unless every connected component of is a path . On the other hand, it would be consistent with our knowledge if both MWIS and -Coloring were polynomial-time solvable on -free graphs for every fixed , however this is currently unknown. Positive results in this direction are limited only to small values of , as explained next.
-free graphs, known also as cographs, are well-understood; in particular, they have bounded cliquewidth, which implies the existence polynomial-time algorithms for all the discussed problems. -free graphs are much more mysterious and only in 2014, Lokshtanov, Vatshelle, and Villanger proposed a novel technique that uses the framework of potential maximal cliques and proved polynomial-time tractability of MWIS in this class . This result was followed by a much more technically complex positive result for -free graphs  and a recent algorithm for Feedback Vertex Set in -free graphs . For coloring with few colors, the state-of-the-art are polynomial-time algorithms for -Coloring of -free graphs  and -Coloring of -free graphs .
In full generality, the so-called Gyárfás’ path argument gives subexponential-time algorithms for both MWIS and -Coloring in -free graphs for any fixed [3, 7, 13]. Using the Gyárfás’ path argument and the three-in-a-tree theorem , it is possible to obtain a quasi-polynomial-time approximation scheme for MWIS in -free graphs whenever every connected component of is a tree with at most three leaves . Note that this covers exactly the cases where NP-hardness is not known. The crucial property of -free graphs that is used in all the works mentioned above is that, due to the aforementioned Gyárfás’ path argument, every -free graph admits a balanced separator consisting of at most closed neighborhoods of a vertex.
The lack of NP-hardness results on one side, and shortage of generic algorithmic tools in -free graphs on the other side, calls for a deeper understanding of the structure of -free graphs for larger values of . In this note we discuss one property that appeared important in the algorithms for MWIS for -free and -free graphs [17, 16, 14], namely the possibiliy to cover a minimal separator with a small number of vertex neighborhoods.
Let be a graph. For a set , a connected component of is a full component to if . A set is a minimal separator if it admits at least two full components. A set is a chordal completion if is chordal (i.e., does not contain an induced subgraph isomorphic to a cycle on at least four vertices). A set is a potential maximal clique (PMC) if there exists an (inclusion-wise) minimal chordal completion of such that is a maximal clique of . Potential maximal cliques and minimal separators are tightly connected: for example, a graph is chordal if and only if every its minimal separator is a clique, and if is a PMC in , then for every connected component of the set is a minimal separator with being one of the full components.
A framework of Bouchitté and Todinca [5, 6], extended by Fomin, Todinca, and Villanger , allows solving multiple computational problems (including MWIS and Feedback Vertex Set) on graph classes where graphs have only a polynomial number of PMCs. While -free graphs do not have this property, the crucial insight of the work of Lokshtanov, Villanger and Vatshelle  allows modifying the framework to work for -free graphs and, with more effort, for -free graphs .
A simple, but crucial in , insight about the structure of -free graphs is the following lemma.
Lemma 1 ().
Let be a -free graph, let be a minimal separator in , and let and be two full components of . Then for every and it holds that .
The above statement is per se false in -free graphs, but the following variant is true and turned out to be pivotal in :
Lemma 2 (, Lemma 20 in the arXiv version).
Let be a -free graph, let be a minimal separator in , and let and be two full components of . Then there exist nonempty sets and such that , , and .
That is, every minimal separator in a -free graph has a dominating set of size at most , contained in the union of two full components of this separator.
In Section 3 we extend the result to -free graphs as follows.
Let be a -free graph and let be a minimal separator in . Then there exists a set of size at most such that .
Section 5 discusses a modified example from  that witnesses that no statement analogous to Theorem 3 can be true in -free graphs. Furthermore, observe that in the statements for -free and -free graphs the dominating set for the separator is guaranteed to be contained in two full components of the separator. This is no longer the case in Theorem 3 for a reason: in Section 5 we show examples of -free graphs where any constant-size dominating set of a minimal separator needs to contain a vertex from the said separator.
The intuition behind the framework of PMCs, particularly visible in the quasi-polynomial-time algorithm for MWIS in -free graphs , is that potential maximal cliques can serve as balanced separators of a graph. Here, is a balanced separator of if every connected component of has at most vertices. The quasi-polynomial-time algorithm of  tried to recursively split the graph into significantly smaller pieces by branching and deleting as large as possible pieces of such a PMC. Motivated by this intuition, in Section 4 we generalize Theorem 3 to dominating potential maximal cliques:
Let be a -free graph and let be a potential maximal clique in . Then there exists a set of size at most such that .
For basic graph notation, we follow the arXiv version of . We outline here only nonstandard notation that is not presented in the introduction.
For a set , by we denote the family of connected components of . A set is complete to a set if every vertex of is adjacent to every vertex of .
Potential maximal cliques.
A set is a potential maximal clique (PMC) if:
none of the connected components of is full to ; and
whenever is a non-edge with , then there is a component such that .
In the second condition, we will say that the component covers the non-edge . As announced in the introduction, we have the following.
Proposition 5 (Theorem 3.15 of ).
For a graph , a vertex subset is a PMC if and only if there exists a minimal chordal completion of such that is a maximal clique in .
We will also need the following statement.
Lemma 6 (cf. Proposition 8 of the arXiv version of ).
For every PMC of and every , the set is a minimal separator.
Let be a graph. A set is a module of if for every . Note that , and all the singletons for are modules; we call these modules trivial. A graph is prime if all its modules are trivial. A module of is strong if and does not overlap with any other module of , i.e., for every module of we have either , or , or .
A partition of is a modular partition of if is a module of for every . The quotient graph is a graph with the vertex set and with if and only if for all and (since and are modules, is an edge either for all pairs and , or for none).
It is well-known (cf. [15, Lemma 2]) that if then the family of (inclusion-wise) maximal strong modules of forms a modular partition of whose quotient graph is either an independent set (if is not connected), a clique (if the complement of is not connected), or a prime graph (otherwise). We denote this modular partition by and we let . For , we abbreviate and with and , respectively.
3 Covering minimal separators in -free graphs
Lemma 7 (Bi-ranking Lemma of , Lemma 17 of the arXiv version).
Suppose is a non-empty finite set and and are two quasi-orders. Suppose further that every pair of two different elements of is comparable either with respect to or with respect to . Then there exists an element such that for every we have either or .
Lemma 8 (Neighborhood Decomposition Lemma of , Lemma 18 of the arXiv version).
Suppose is a graph and is subset of vertices such that and is connected. Suppose further that vertices respectively belong to different elements of the modular partition such that and are adjacent in the quotient graph . Then, for each vertex at least one of the following conditions holds:
there exists an induced in such that is one of its endpoints, while the other three vertices belong to ;
is a clique and the neighborhood of in is the union of some collection of maximal strong modules in .
In particular, if is not a clique, then the last condition cannot hold.
Let be a -free graph, let be a minimal separator in , and let and be the vertex sets of two full components of . If for some , then we are done by setting , so assume . For each , fix two different maximal strong modules and of that are adjacent in . Furthermore, pick arbitrary and .
For each , we apply Lemma 8 to and . We say that a vertex is of type if . We say that a vertex is of type if is not of type and there is an induced in with being one of the endpoints and the other three vertices belonging to . Finally, we say that a vertex is of type if is neither of type nor . Lemma 8 asserts that if there are vertices of type , then is a clique and the neighborhood in of every vertex of this type is the union of a collection of maximal strong modules of . For , let be the set of vertices that are of type and .
We need the following claim.
Let and let be of type . Then is complete to .
By Lemma 8, is a clique and both and are the unions of some disjoint collections of maximal strong modules of . The claim follows.
Since is -free, . Furthermore, if we set , then
In the rest of the proof, we construct sets and such that for . We will conclude that satisfies the statement of the lemma, because and we will ensure that and .
We start with constructing the set . If , then we set . Otherwise, let be a vertex with inclusion-wise minimal set . Furthermore, let be an arbitrary neighbor of in ; exists since is a full component of . Also, let be vertices of an induced with ; recall here that is of type . We set and claim that .
Assume the contrary, and let . By the choice of and since , there exists . By Claim 1, . Then, is an induced in , a contradiction.
Hence, we constructed of size at most such that . A symmetric reasoning yields of size at most such that .
We are left with constructing . If , then we take and conclude. In the remaining case, is non-empty, so both and are cliques.
For each , we define a quasi-order on as follows. For , if . An unordered pair is a butterfly if and are incomparable both in and in , that is, if each of the following four sets is nonempty:
See Figure 1 for an illustration.
Lemma 7 allows us to easily dominate subsets of that do not contain any butterflies:
Let be such that there is no butterfly with . Then there exist a vertex and such that .
If , the claim is trivial, so assume otherwise. Let us focus on quasi-orders and , restricted to . Since there are no butterflies in , the prerequisities of Lemma 7 are satisfied for and . Hence, there exists with or for every . For , let be an arbitrary neighbor of in (it exists as is a full component of ). For every , there exists such that , hence . We conclude that , as desired.
If there is no butterfly at all, then we apply Claim 2 to , obtaining vertices and set . Thus, we are left with the case where at least one butterfly exists.
Let be a butterfly with inclusion-wise minimal set
Furthermore, pick the following four vertices
Claim 1 ensures that and .
Let . We claim the following:
There is no butterfly with .
Assume the contrary, and let be a butterfly with . By the minimality of , as but , there exists . By symmetry, assume that and ; see Figure 2.
By Claim 1, and . If , then would be an induced in . Otherwise, if , then would be an induced in . As in both cases we have obtained a contradiction, this finishes the proof.
4 Covering PMCs in -free graphs
Let be a -free graph and let be a potential maximal clique in . Then there exists a set of size at most and a set of size at most such that
Let be an inclusion-wise minimal set of components of such that for every nonedge in there exists a component that covers .
If , then is a clique in and thus we can put and for an arbitrary . Otherwise, pick any . By the minimality of , there exists a nonedge in that is covered by and by no other component of .
Assume that there is no component with . Then , so . Hence, we can set and . Symmetrically, we are done if there is no component with .
In the remaining case, pick arbitrary components with and with . Since is the only component of that covers , we have and ; in particular, . We claim that we can set and . That is, we claim that
Assume the contrary, and let be such that , , , , and .
Since is a nonedge of , there exists that covers . Similarly, there exists that covers . By the choice of , we have . Further, since is the only component of covering , and ; in particular, . See Figure 3.
Let be an arbitrary neighbor of in , let be an arbitrary neighbor of in , let be a shortest path from to with all internal vertices in , and let be a shortest path from to with all internal vertices in . Then, is an induced path with at least vertices, a contradiction. This proves (1) and concludes the proof of Lemma 9. ∎
In this section we discuss two examples showing tightness of the statement of Theorem 3: we show that it cannot be generalized to -free graphs and that a small dominating set of a minimal separator may need to contain elements of the said separator. The examples are modifications of a corresponding example presented in the conclusions of .
Consider the following graph . We create three sets of vertices each, , , and . We set . For the edge set of , we turn and into cliques and add edges and , for all . This concludes the description of the graph ; see Figure 4. Note that is a minimal separator in with and being two full components of .
First, note that for every , . Thus, any set dominating has to contain at least vertices.
Second, note that is -free. To see this, let be an induced path in . Since and are cliques, contains at most two vertices from each , , and these vertices are consecutive on . Since is an independent set in , cannot contain more than one vertex of in a row. Hence, contains at most three vertices of . Consequently , as desired. Note that if , then there is an induced in , for example .
Here, let us modify the graph from the first example by turning into a clique. Still, is a minimal separator in with and being two full components of .
First, note that for every , we still have . Thus, any set dominating that is disjoint with has to contain at least vertices.
Second, note that is -free. To see this, observe that can be partitioned into three cliques, , , and , and any induced path in contains at most two vertices from each of the cliques. Note that if , then there is an induced in , for example .
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