Given a fixed forbidden graph , what is the minimum number of edges that any graph on vertices can have such that contains no copy of , but the addition of any single edge to results in a copy of ? This question is a variation of the well-known forbidden subgraph problem in extremal graph theory, which asks for the maximum number of edges in an -free graph on vertices. Asking for the minimum number of edges instead (and tailoring the definition so that this is a sensible question) yields the notion of the saturation number of a graph , first defined by Erdős, Hajnal, and Moon , albeit with slightly different terminology.
Let be a graph. For any graph , we say that is -free if it contains no subgraph isomorphic to . We say that is -saturated if it is -free and for any , the graph contains a subgraph isomorphic to . For , let denote the set of all -saturated graphs on vertices, and let the saturation number of be
In the event that , we adopt the convention that . Note that this will only happen if has no edges.
In their paper introducing the concept, Erdős, Hajnal, and Moon  determined in the case where is a complete graph. Since then, the saturation numbers have been determined for various classes of graphs. A nice survey on these results and more was written by Faudree, Faudree, and Schmitt .
The best known general upper bound on was given by Kászonyi and Tuza  and later slightly improved by Faudree and Gould . However, as mentioned in  and in , there is no known nontrivial general lower bound on this function. In this paper we give such a bound, and determine a class of graphs for which this bound is asymptotically sharp: for such graphs, we can prove that , where and the term depend on only . (We remark that it is not known, in general, that the limit even exists, even though it is known  that is always bounded by a linear function of ; the existence of this limit was stated as a conjecture by Tuza .)
This class of graphs includes all threshold graphs as well as some non-threshold graphs. In particular, many previously-studied classes of graphs fall into this class, including cliques , stars , generalized books , disjoint unions of cliques , generalized friendship graphs , and several of the “nearly complete” graphs of . Our result can be considered as an asymptotic common generalization of these previous results: at the cost of no longer determining the exact saturation number as in the previous results, we obtain a simple unified proof that gives the saturation number up to an additive constant number of edges.
The rest of the paper is organized as follows. In Section 2 we state and prove our general lower bound on the saturation number, and prove an upper bound on the saturation number of the graph obtained from a graph by adding a dominating vertex. In Section 3, we define the sat-sharp graphs to be the graphs whose saturation numbers are asymptotically equal to the lower bound of Section 2, and prove that this class of graphs is closed under adding isolated vertices and dominating vertices. In Section 4 we discuss threshold graphs, which are contained within the class of sat-sharp graphs and encompass several graphs whose saturation numbers were previously determined. Finally, in Section 5, we prove that any graph consisting of a disjoint union of cliques is sat-sharp, and discuss the implications of this.
2. A weight function and some general bounds
In this section, we will define a weight function for a general graph , and prove that it gives a lower bound on the saturation number . We will also prove a general bound relating the saturation number of to the saturation number of the graph obtained from by adding a dominating vertex.
For a vertex in a graph , let and denote the open and closed neighborhoods of respectively:
Let denote the degree of , and for a vertex set , let denote the number of neighbors of in the set :
When the graph is clear from context, we omit it from our notation and simply write , , or as appropriate.
Let be an edge in a graph with . Define the weight of the edge by
Define the weight of the graph by
If , we define .
For every graph , there exists a constant such that
First observe for all and that the claim is trivial when , so (as is an integer) we may assume that . Let be an -saturated graph, let be a vertex of minimum degree in , and let , so that . Observe that if , then the degree-sum formula immediately gives , so we may assume that . As both of these quantities are integers, we have .
Consider any vertex . By hypothesis, the graph contains a copy of . Let be an embedding of into . Since is -free, the new edge must be the image of some edge . We may take our notation so that . Let and let , so that .
We first claim that ; this requires considering two cases, depending on whether or . If , then
Similarly, if , then . This establishes the claim.
Now, observe that regardless of whether or , we have
So in , the vertex has guaranteed neighbors in , together with at least additional edges which may go to or may go to , where . Therefore,
for all .
Now, note that . So it follows that
Since we assume , the value , considered as a formal function of the quantity , is maximized at whenever (the case , which would imply that this formal function has a negative derivative in , implies that ). Therefore,
for . ∎
A central goal of this paper is to explore the effect on the saturation number of the operation of adding a dominating vertex to , as shown in Figure 1. It turns out that that this gives a general upper bound on the saturation number of the new graph in terms of the saturation number of ; we wish to know when this upper bound is sharp. We believe that this upper bound is in the same general spirit as Lemma 9 of Kászonyi–Tuza .
If is obtained from by adding a dominating vertex , then for all , we have .
It suffices to produce an -saturated graph with at most the indicated number of edges. Let be a minimum -saturated graph on vertices, and let be obtained from by adding a new dominating vertex . It is clear that ; we show that is -saturated.
First we argue that is -free. Suppose to the contrary that is an embedding of into . If , then is an embedding of into . Hence has a copy of and thus a copy of , contradicting the -saturation of . If , then the restriction of to is an embedding of into , again a contradiction.
Hence we can assume that and there is some vertex with . Construct a new embedding by letting and taking for all . Since dominates the image of in (as was a dominating vertex of ), we see that is still a valid embedding. Hence we have again obtained a copy of in , a contradiction. We conclude that is -free.
Finally we argue that adding any missing edge to produces a new copy of . Since is dominating, any missing edge in is an edge of the form where . Now contains a copy of , since is -saturated; adding the dominating vertex to this copy of gives a copy of in . ∎
3. Sat-sharp graphs
For a graph , let , provided that this limit exists. Say a graph is sat-sharp if . Moreover, say that a graph is strongly sat-sharp if . Note that any strongly sat-sharp graph is also sat-sharp. Also, note that by adopting the convention that when , we can conclude that any graph with no edges is strongly sat-sharp since for all .
In this section we will show that the classes of sat-sharp graphs and strongly sat-sharp graphs are each closed under adding isolated and dominating vertices. To express these results concisely, we write statements like “if is (strongly) sat-sharp, then is (strongly) sat-sharp” as shorthand for the pair of statements “if is sat-sharp, then is sat-sharp; if is strongly sat-sharp, then is strongly sat-sharp”.
As is strongly sat-sharp, these closure results immediately imply that all threshold graphs are strongly sat-sharp (as we will discuss in Section 4). They also imply that any graph which can be proven to be (strongly) sat-sharp gives rise to many (strongly) sat-sharp graphs derived from by these operations. In particular, we will prove in Section 5 that a disjoint union of cliques is strongly sat-sharp, although it is not in general a threshold graph; this implies that any graph obtained from a disjoint union of cliques via these operations is also strongly sat-sharp.
If is a (strongly) sat-sharp graph, and is obtained from by adding isolated vertices, then is (strongly) sat-sharp, and .
For all , a graph is -saturated if and only if it is -saturated, hence for all sufficiently large . ∎
To handle the operation of adding a dominating vertex, we prove the following two lemmas, which taken together show that the class of (strongly) sat-sharp graphs is closed under the operation of adding a dominating vertex.
Let be a -vertex (strongly) sat-sharp graph, and let be obtained from by adding a dominating vertex . If has no isolated vertices, or if , then and . Moreover, is also (strongly) sat-sharp.
Let be a -vertex graph with an isolated vertex , and let be obtained from by adding a dominating vertex . If , then is strongly sat-sharp, with and .
Note that Lemma 8 does not actually require the graph to be sat-sharp, although that is the main case we are concerned with. In the case where has no edges and so , the hypothesis of Lemma 8 applies, yielding ; this is an asymptotic version of the exact result of Kászonyi and Tuza .
Proof of Lemma 7.
Let , so that when is sat-sharp and when is strongly sat-sharp.
By Lemma 5, we have
If we can prove that , then Lemma 4 will give
In particular, this implies that and that , so if is (strongly) sat-sharp, then is (strongly) sat-sharp.
An edge can be viewed (and its weight computed) either as an edge of or as an edge of . We will use and to refer to the weight of such an edge computed in or , respectively. Observe that if , then when we pass to , we add as a new element of and change nothing else about the sets or . Hence, for all .
The only remaining edges of are edges of the form for . We claim that all such edges have weight at most . If is isolated in , then we have
where the last inequality follows from the assumption that (since we assumed that is isolated and that either obeys this weight inequality or is isolate-free).
On the other hand, if is not isolated in , let be another edge incident to . Observe that
and that for any edge . It follows that . Hence, an edge of minimum weight in is found among the edges of , and the smallest such weight is . ∎
Proof of Lemma 8.
We again write to refer to the weight of an edge computed in and to refer to the weight of an edge when computed in .
As previously discussed, we have for every edge . On the other hand, considering the isolated vertex , we see that , as and .
If , then this implies , with the only edges of minimum weight being those edges joining with an isolated vertex of . This establishes the first claim of the lemma.
Lemma 4 now gives the lower bound
We establish a matching upper bound by constructing an -saturated graph on vertices, for any .
Let any be given, and write , where . Let be the -vertex graph consisting of disjoint copies of and a single copy of . Clearly
So if we can argue that is -saturated, then we will have , and we will have that is strongly sat-sharp.
It is clear that is -free, since is connected and has vertices, while every component of has at most vertices. We claim that adding any edge to produces a subgraph isomorphic to . Let be a missing edge in ; we may assume that lies in a copy of . Let be the set of vertices of the copy of containing .
Now observe that we can embed into by any injection that satisfies:
and with vertices in , there is enough room to complete the last part of the embedding. The key point is that there is no edge, in , from to any vertex of except for , and all vertices of except for are being embedded into a clique of , so any edges they require are present. Thus, is -saturated, which completes the proof. ∎
4. Threshold Graphs
A natural class of strongly sat-sharp graphs is the class of threshold graphs. A simple graph with vertex set is a threshold graph if there exist weights such that, for all , we have if and only if . Threshold graphs were first introduced by Chvátal and Hammer [2, 3], albeit with a slightly different definition than the one we give here.
Threshold graphs admit many equivalent characterizations. For our purposes, the following characterization is the most useful one.
For a simple graph , the following are equivalent:
is a threshold graph;
can be obtained from by iteratively adding a new vertex which is either an isolated vertex, or dominates all previous vertices;
In fact,  gives several other equivalent characterizations of threshold graphs, but this is the one we will be interested in. The results of Section 3, together with this characterization, immediately imply that all threshold graphs are strongly sat-sharp. Furthermore, when a construction sequence for a threshold graph is given, one can use the lemmas from Section 3 to easily compute by iteratively computing the weight of each intermediate subgraph, keeping track of the previous subgraph’s weight and whether or not it had an isolated vertex.
As discussed in the introduction, several graphs whose saturation numbers were determined in previous work fall into the class of strongly sat-sharp graphs. In particular, complete graphs , stars , generalized books , stars plus an edge , and “nearly complete” graphs of the form for are all threshold graphs. Thus, all of these graphs are strongly sat-sharp, and their saturation number is determined (up to a constant number of edges) by the results of Section 3.
Kászonyi and Tuza  observed the “irregularity” that if is the graph obtained from by adding a pendant edge, then while , so that for sufficiently large even though . Both and the graph are threshold graphs; in terms of our weight function, the irregularity can be seen to arise from the fact that all edges of have weight while the pendant edge of has weight .
5. -saturated construction when is the disjoint union of cliques
Faudree, Ferrara, Gould, and Jacobson  determined the saturation numbers of generalized friendship graphs , consisting of copies of which all intersect in a common but are otherwise pairwise disjoint. When , this includes the case of , consisting of disjoint copies of . They also determined the saturation numbers of two disjoint cliques, , when , but left determining the saturation number of three or more disjoint cliques with general orders as an open problem. Here, we give a proof that all of these graphs are strongly sat-sharp, and determine their saturation numbers up to an additive constant for all sufficiently large .
Let be positive integers. The graph is strongly sat-sharp. In particular,
for some constants depending only on and for all .
First, note that . So by Lemma 4,
for some constant .
On the other hand, let be the graph on vertices defined as the join, where , the disjoint union of a clique on vertices and a set with isolated vertices. Figure 2 shows the graph that is constructed for .
We claim that is -free and -saturated. To see that is -free, consider its maximal cliques. Let denote the subgraph of induced by the vertices of the and the . Then is a maximal clique with vertices. All other maximal cliques of are formed from the and one vertex from . Therefore, if we were to find a copy of in , then each of the disjoint cliques of must be found in . But only has vertices so this cannot happen.
To see that is -saturated, consider the graph for some . Without loss of generality, either or and . In either case, the vertices of form a -clique in , while at least vertices of the remain disjoint from this clique and can be used to embed the remaining cliques of . So is -saturated.
Since has edges, it follows that
for some constant . Therefore, is strongly sat-sharp. ∎
An immediate corollary to this proposition and the results of Section 3 is the following result.
Let and be positive integers. Let , and let . Then is sat-sharp. In particular,
for some constants depending only on and for all .
Note that this class of graphs includes all generalized friendship graphs for . Since for is a threshold graph, we already know from the discussion in Section 4 that it is strongly sat-sharp.
While a disjoint union of cliques is not, in general, a threshold graph, each of its components is a threshold graph. Proposition 10 therefore suggests that perhaps a disjoint union of threshold graphs is always strongly sat-sharp. More boldly, the following conjecture appears to be plausible:
If and are (strongly) sat-sharp graphs, then their disjoint union is (strongly) sat-sharp. That is, the class of (strongly) sat-sharp graphs is closed under taking disjoint unions.
Conjecture 12, together with the other closure properties from Section 3, would immediately imply Proposition 10. We have found ad-hoc constructions for some small disjoint unions of particular threshold graphs which suggest that Conjecture 12 might hold, but it has been difficult to extract a general construction.
We would like to thank Ron Gould for the interesting talk on saturation numbers that he gave at the Atlanta Lecture Series in 2018 which stimulated work on this paper.
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