For a graph , a set is a minimal separator if there are at least two connected components of with (so that is an inclusion-wise minimal set that separates a vertex of from a vertex of ). Around the year 2000, Bouchitté and Todinca presented a theory of minimal separators and related objects called potential maximal cliques and showed their usefulness for providing efficient algorithms . In particular, the Maximum Weight Independent Set problem (given a vertex-weighted graph, find a subset of pairwise nonadjacent vertices of maximum total weight) can be solved in time bounded polynomially in the size and the number of minimal separators in the graph. This result has been generalized by Fomin, Todinca, and Villanger to a large range of problems that can be defined as finding an induced subgraph of constant treewidth with some CMSO-expressible property ; this includes, for example, Longest Induced Path or Max Induced Forest, which is by complementation equivalent to Feedback Vertex Set.
When do these metaalgorithmic results give efficient algorithms? In other words, which restrictions on graphs guarantee a small number of minimal separators? On one hand, it is easy to see that an -vertex chordal graph has minimal separators. On the other hand, consider the following two negative examples. For , the -prism consists of two -vertex cliques with vertex sets and and a perfect matching . It is easy to see that the -prism has minimal separators: any choice of one endpoint of each edge gives a minimal separator, except for the choices and . The -theta consists of independent edges , a vertex adjacent to all vertices and a vertex adjacent to all vertices (the intuition behind the notation is that the graph consists of paths of length 3, joining and ). Again, any choice of one endpoint of each edge gives a minimal separator. Thus, both the -prism and the -theta have an exponential (in the number of vertices) number of minimal separators.
In 2019, Milanič and Pivač initiated a systematic study of the question which graph classes admit a small bound on the number of minimal separators in its members [5, 6]. A graph class is tame if there exists a polynomial such that for every the number of minimal separators of is bounded by . Clearly, if is tame, then Maximum Weight Independent Set and all problems captured by the formalism of  are solvable in polynomial time when the input graph comes from . On the opposite side of the spectrum, is feral if there exists such that for infinitely many graphs it holds that has at least minimal separators. Following the previous examples, the class of chordal graphs is tame while the class of all -prisms and/or all -thetas (over all ) is feral. Milanič and Pivač provided a full tame/feral dichotomy for hereditary graph classes (i.e., closed under vertex deletion) defined by minimal forbidded induced subgraphs on at most vertices [5, 6].
A subsequent work of Abrishami, Chudnovsky, Dibek, Thomassé, Trotignon, and Vuskovič  indicated that the main line of distinction between tame and feral graph classes should lie around the notion of a -creature. A -creature in a graph is a tuple of pairwise disjoint nonempty vertex sets such that (i) and are connected, (ii) is anti-adjacent to and is anti-adjacent to , (iii) every has a neighbor in and every has a neighbor in ; (iv) and and can be enumerated as , such that if and only if . We say that is -creature-free if does not contain a -creature as an induced subgraph. Similarly as in the examples of the -prism and the -theta, any choice of one endpoint of every edge gives a minimal separator in the subgraph induced by the creature (which, in turn, can be easily lifted to a minimal separator in ). Hence, if contains a -creature as an induced subgraph, it contains at least minimal separators. In fact, the notion of a -creature is a common generalization of the examples of the -prism and the -theta. Indeed, the -theta contains a -creature with and while the -prism contains a -creature with , , , and . In particular, Abrishami et al. conjectured that if for a hereditary graph class there exists such that no contains a -creature as an induced subgraph, then is tame. (Observe that a presence of arbitrarily large creatures in a hereditary graph class does not immediately imply that the graph class is feral, as the sets and can be of superpolynomial size in .)
A counterexample to the conjecture of  has been provided by Gartland and Lokshtanov in the form of a -twisted ladder . They observed that, despite the fact that the conjecture of  is false, every example they can construct “looks like a twisted ladder”, which indicates that the tame/feral boundary for hereditary graph classes should not be far from the said conjecture. To support this intuition, they introduced the notion of a -skinny ladder (a graph consisting of two induced antiadjacent paths , , and independent set , and edges ), noted that a -skinny-ladder is an induced minor of every counterexample they constructed, and proved the following.
For every there exists a constant such that if a graph is -creature-free and does not contain a -skinny-ladder as an induced minor, then the number of minimal separators in is bounded by , that is, quasi-polynomially in the size of .
Gartland and Lokshtanov conjectured that this dependency should be in fact polynomial. Our main result of this paper is a proof of this conjecture.
For every there exists a polynomial of degree such that every graph that is -creature-free and does not contain -skinny-ladder as an induced minor contains at most minimal separators.
That is, every hereditary graph class for which there exists such that no member of contains a -creature nor -skinny-ladder as an induced minor, is tame.
As proven in , theorem 2 implies a dichotomy result into tame and feral graph classes for all hereditary graph classes defined by a finite list of forbidden induced subgraphs. (For the exact definitions of graphs in the statement, we refer to .)
Let be a graph class defined by a finite number of forbidden induced subgraphs. If there exists a natural number such that does not contain all -theta, -prism, -pyramid, -ladder-theta, -ladder-prism, -claw, and -paw graphs, then is tame. Otherwise is feral.
That is, we want a set of maximum possible size that is not only independent, but no vertex outside is adjacent to more than one vertex of . In the proof of theorem 1 of , an important step is to prove that a minimal separator with huge gives rise to a large skinny ladder as an induced minor. Our main technical contribution is an improved way of analysing minimal separators with small .
For every there exists a polynomial of degree , such that the following holds. For every -creature-free graph , the number of minimal separators satisfying is at most .
Let be a graph, be a vertex of , and be a subset of vertices. By we denote the set of neighbors of . Similarly, by we denote the set . If the graph is clear from the context, we simply write and .
For sets , whenever we write , the set difference operation associates from the left, meaning that is equivalent to (and, alternatively, to ).
By we denote the graph obtained from by deleting all vertices from along with incident edges, and by we denote the graph induced by the set , i.e., . By we denote the set of connected components of , given as vertex sets.
A matching in is a set of pairwise disjoint edges. We say that a matching is a semi-induced matching between and if for all , if and only if .
For vertices , a set is a --separator if and are in different connected components of . We say that is a minimal --separator if it is a --separator and no proper subset of is a --separator. A set is a minimal separator if it is a minimal --separator for some . Equivalently, is a minimal separator if there are at least two components such that . Any component with is called full to ; a minimal separator has at least two full components.
The following result of Gartland and Lokshtanov will be a crucial tool used in our argument.
Lemma 5 (Gartland and Lokshtanov ).
If is a -creature-free graph, then for every it holds that .
Let us also recall the crucial definition. For a set we define
3 Proof of theorem 4
We prove the theorem by induction on with the exact bound of minimal separators.
Note that if , then , since for any , the set satisfies the required properties. Thus, in the base case, when , the only candidate for is the empty set, therefore the claim holds vacuously. Also, the claim is immediate for , so we assume .
Let be a minimal separator of , and let and be two connected components of that are full to . If there is a vertex such that , then . There are at most such separators by lemma 5; we may therefore assume that no such vertex exists. Let be a minimal connected subset of that still dominates , i.e. such that . Let be such that is still connected. Such a vertex can be found, for instance, as a leaf of a spanning tree of . We define the following sets that will be important throughout the proof, see fig. 1.
We let . In words, is a private neighbor (with respect to ) of in . Such a vertex exists by the minimality of .
. That is, contains the vertices of that have a common neighbor with in .
. Similarly, contains the vertices of that have a common neighbor with in .
Our goal is now to identify a small set that dominates . We will repeatedly use lemma 5 on the vertices of this set in order to bound the number of choices for . We then show that we can find a minimal separator in such that is a full component in and there is a component containing . We will be able to show that which allows us to conclude using the induction hypothesis on .
Let be a minimal set such that . Then contains a -creature.
Proof of Claim.
By the minimality of , each vertex of has a private neighbor in . Hence, there is a semi-induced matching between and a size- subset of , say . We obtain the -creature by considering the sets , see fig. 1(a). Indeed, note that by our choice of , we have that is connected; has no neighbors in , as it is a private neighbor of ; has no neighbors in since ; clearly there are no edges between and since is a separator; dominates and dominates .
Let be a minimal set of such components that dominates . Then contains a -creature.
Proof of Claim.
By the minimality of , for each there exists a vertex that is dominated only by vertices of . Let be a vertex of that is adjacent to and that is closest to in . Note that the edges form a semi-induced matching between and in . Let be the set of internal vertices on a shortest path between and via . Note that and that is anti-adjacent to . Indeed, the vertices of are not adjacent to , as this would contradict the minimality of the distance between and ; and for any , the vertices of are not adjacent to as this vertex is only dominated by vertices of , by our choice of . Then we obtain a -creature by considering the sets , see fig. 1(b). Indeed, note that and are connected; there are no edges between and since , neither edges between and as mentioned above. Note also that is anti-adjacent to as argued in the proof of Claim 1. Finally, note that dominates , since every either has a neighbor in , or in if . Finally, it is easy to see that dominates .
Let be as in Claim 2. For each , let be a minimal set that dominates . Then contains a -creature.
Proof of Claim.
By the minimality of , there is a semi-induced matching between and a size- subset of . We obtain a creature by considering the sets , see fig. 1(c). Indeed, note that is connected by definition; is not adjacent to since and is not adjacent to since is a connected component of ; is not adjacent to since is a separator; dominates by definition of and .
Let , where , , and for are as defined in Claims 3, 2 and 1, respectively. For all , let . Let . Note that contains since , that contains since , and that contains . The latter is due to the fact that the vertices in dominate by choice, and each where dominates . We illustrate this situation in fig. 3. It remains to get a grip on .
To do so, let , and note that separates from . Let be a minimal subset of that still separates from . Note that there are components and that are full to and is a minimal separator. Now let , so there are at most choices for , by lemma 5. We observe that , which is due to the fact that dominates , and that separates from .
There are at most choices for , and at most choices for .
Proof of Claim.
Now, let and . Note that . Moreover, is a connected component of , and there is a connected component of that contains . We conclude that is a minimal separator of , with and being connected components of that are full to . We now show that we can use the induction hypothesis to bound the number of choices for .
Proof of Claim.
Let be an independent set such that for all , . Let ; is still an independent set since . We argue that for all , . Suppose that . Since , we have that and therefore . We may now assume that . Suppose that . Since , we conclude that , otherwise would have at least two neighbors in , a contradiction with the choice of in in the graph . This means that , and therefore , which is a contradiction with our assumption that .
The number of choices for , , , and is at most , see Claim 4. For , there are at most choices by Claim 5 and the induction hypothesis. Given , , and , there are at most choices for and we obtain as . Taking into account also at most separators for which there exists with , the number of separators of is bounded by
This completes the proof.
4 Wrapping up the proof of theorem 2
Lemma 6 ().
If is a -creature-free graph that contains a minimal separator with , then contains a -skinny-ladder as an induced minor.
Let and be as in the lemma statement. Let be an independent set of size such that no vertex is adjacent to more than one vertex of .
Let and be two full sides of . Lemma 9 of  asserts that there exists an induced path in , an induced path in , and a set of size at least such that dominates and dominates .
This is exactly the situation at the end of the first paragraph of the proof of Lemma 15 of . A careful inspection of that proof shows that the remainder of the proof (as well as the invoked Lemmata 8, 13 and 14) do not use other assumptions of Lemma 15. Hence, we obtain the conclusion: a -skinny ladder as an induced minor of .
In theorem 2 we showed that if a graph class excludes -creatures as induced subgraphs and -skinny ladders as induced minors, then is tame. However, note that while -creatures have exponential (in ) number of minimal separators, this is not the case for -skinny ladders: the class of -skinny ladders (over all ) is tame. Thus the implication reverse to the one in theorem 2 does not hold.
Observe that the full tame/feral dichotomy for arbitrary hereditary graph classes is simply false due to some very obscure examples. Let be the -theta graph: paths of length with common endpoints. Note that has minimal separators ( of them choose one internal vertex on each path) and vertices, so the number of minimal separators of is around . Hence, the hereditary class of all induced subgraphs of all graphs for is neither tame nor feral.
However, it is still interesting to try to obtain a tighter classification between tame and feral graph classes for some more “well-behaved” hereditary graph classes. As discussed in Conjecture 4 of , a good restriction that excludes artificial examples as in the previous paragraph is to focus on induced-minor-closed graph classes.
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