On overlapping Feynman (sub)graphs

We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph. This, in turn, allows us to derive"non-overlap"theorems for one-particle-irreducible subgraphs with $2$, $3$ and $4$ external legs. As an application, we provide a simple justification of the skeleton expansion for vertex functions with more than five legs, in the case of scalar field theories. We also discuss how the skeleton expansion can be extended to other classes of graphs.

then extend this result to other classes of functions, including the high enough derivatives of the Luttinger-Ward functional.

II. GRAPHS AND SUBGRAPHS
In perturbative calculations, quantities are computed by summing Feynman graphs made of two basic elements: free propagators that are represented graphically as lines, and vertices that are represented by points with a certain number of legs. 1 We stress that vertex legs are not to be seen as lines, but rather as little anchors on which lines can be attached (or not). In what follows, we introduce more precisely the notion of graph together with some related concepts. In particular, we describe how a subgraph of a graph is inserted within that graph by means of both connecting lines and returning lines. This will then allow us to describe all possible overlaps between subgraphs of a given graph.

A. Graphs
We define a graph G as any collection of vertices and lines with the property that the two ends of any line of G are attached to vertices of G. We can distinguish two types of vertices within the graph: those whose legs are all connected to lines of G are called internal vertices, while the others are called external (or boundary) vertices. The legs of external vertices are of two types: legs attached to lines of G and legs attached to no line. We call the latter the external legs of the graph G and denote them as n ext (G) in the following.
We stress that our definition of graph excludes the possibility of lines with one end not attached to a vertex. This is just a convenient choice for the subsequent discussion, and, if needed, we can always attach such free lines to the external legs of a graph. Reciprocally, any graph including such free lines is associated to a unique graph that has no external lines. We also exclude lines which are not connected to any vertex. These are just trivial elements (disconnected from the rest) that can again be added at will when needed. There are no other restrictions for the moment, so the graphs could be one-particle-reducible, unamputated or even disconnected. Restrictions will be considered when appropriate. one particular subgraph with six external legs as well, e 1 , e 2 and e 3 , . . . , e 6 . We have chosen a oneparticle-irreducible graph for illustration but the discussion in Secs. II and III applies to any type of graph as defined in Sec. II. The leg labels e 1 and e 2 have been introduced for later purpose, see Sec. III.
In Fig. 1, we draw one example of graph in ϕ 4 theory. We shall use it recurrently to illustrate the various notions to be introduced below.

B. Subgraphs
A subgraphḠ of a graph G is any collection of vertices and lines of G that forms a graph in the sense defined above. We write this asḠ ⊂ G. 2 We mention that any internal vertex ofḠ is necessarily an internal vertex of G. In contrast, an external vertex ofḠ can be either an external vertex of G or an internal vertex of G.
Let us also mention that, when seen as a part of G, the legs of the external vertices of G are now of three different types: legs attached to lines ofḠ, legs attached to lines of G that are not inḠ and legs attached to no line (and thus corresponding to external legs of the original graph G). Among these three types of legs attached to the external vertices, we refer to the last two as the external legs of the subgraphḠ. It is clear that the subgraph can be made a separate entity, disconnected from the original graph, by cutting all lines that are attached to its external legs. Indeed, these are the only lines that connect a vertex of the subgraph to a vertex in the rest of the graph. 3 An example of subgraph is shown in Fig. 1. We see clearly what are the external vertices, and thus the external lines that need to be cut to make the subgraph disconnected from the original graph (these are the lines connected to the legs e 3 , . . . , e 6 ).

C. Connecting lines and returning lines
The subgraphḠ is said to be dense within G if its vertices exhaust all vertices of G. In the opposite case, we can define a new subgraph, known as the complementary graph of G within G, denoted G/Ḡ and formed by all the remaining vertices and all the lines that connect them. Together with the vertices ofḠ, the vertices of G/Ḡ exhaust all the vertices of G. This is not so, however, for the lines. Indeed, there might be lines that connect one vertex ofḠ and one vertex of G/Ḡ and which, therefore, do not belong neither toḠ nor to G/Ḡ. We call these lines the connecting lines ofḠ within G. Obviously, these can also be seen as the connecting lines of G/Ḡ within G.
There might also be certain lines that connect vertices ofḠ but which do not belong toḠ.
We call these returning lines ofḠ within G. complementary subgraph G/Ḡ could be disconnected fromḠ. The equivalence works in the case of a connected graph G though.
The notions of connecting and returning lines provide a graphical representation of how a given subgraphḠ is inserted within a graph G, see Fig. 2. This structure will be central in the following developments. If we take the example of Fig. 1, we see that the considered subgraph has one returning line (the thin line connected to the legs e 5 and e 6 ) and two connecting lines (the two thin lines attached respectively to the legs e 3 and e 4 ). The remaining lines and vertices (in the bottom right of the figure) form the complementary graph.

III. OVERLAPPING SUBGRAPHS
We are now ready to discuss how two subgraphsḠ 1 andḠ 2 of a given graph G can overlap with each other. In fact, for the moment, the original graph G will play no role and we can equally think in terms of the overlap of two original graphs. Overlap between two graphsḠ 1 andḠ 2 . The overlap graphC =Ḡ 1 ∩Ḡ 2 is the (maximal) common subgraph ofḠ 1 andḠ 2 . This common subgraph has n c i connecting lines withinḠ i and n r i returning lines withinḠ i . We denote by x, x 1 and x 2 the numbers of external legs ofC,Ḡ 1 /C andḠ 2 /C that are connected neither to connecting lines nor to returning lines.
By overlapping subgraphs or graphs, we mean thatḠ 1 andḠ 2 have certain vertices and lines in common. We shall in fact consider the collection of all common vertices and lines betweenḠ 1 andḠ 2 . It is quite obvious that, if a line is common toḠ 1 andḠ 2 , then the two vertices attached to its ends are also common toḠ 1 andḠ 2 . It follows that this common collection of lines and vertices forms a graph, referred to as the overlap graph betweenḠ 1 andḠ 2 , which we denoteḠ 1 ∩Ḡ 2 in what follows.

A. Overlap pattern
This common graphḠ 1 ∩Ḡ 2 is in fact a subgraph of bothḠ 1 andḠ 2 . We can then apply the results of the previous section twice and introduce two sets of connecting lines, n c 1 and n c 2 in number, as well as two sets of returning lines, n r 1 and n r 2 in number. This leads to the graphical representation shown in Fig. 3, where, for later use, we have also introduced the numbers of external legs ofḠ 1 ∩Ḡ 2 ,Ḡ 1 /(Ḡ 1 ∩Ḡ 2 ) andḠ 2 /(Ḡ 1 ∩Ḡ 2 ) attached neither to connecting lines nor to returning lines, and denoted respectively as x, x 1 and x 2 . It is important to stress that no connecting line ofḠ 1 ∩Ḡ 2 withinḠ 1 can be a connecting line of G 1 ∩Ḡ 2 withinḠ 2 , or vice-versa. Otherwise, this line would be a line of bothḠ 1 andḠ 2 and then an element ofḠ 1 ∩Ḡ 2 , that is not a connecting line. The same remark applies to the returning lines.

B. Counting external legs
The external legs ofḠ 1 are those labelled x 1 , x as well as the n t 2 ≡ n c 2 +2n r 2 legs attached to the connecting and returning lines ofC withinḠ 2 . Reciprocally, the external legs ofḠ 2 are those labelled x 2 , x as well as the n t 1 ≡ n c 1 + 2n r 1 legs attached to the connecting and returning lines ofC withinḠ 1 . We can then write On the other hand, the number of external legs ofḠ 1 ∩Ḡ 2 is Finally, it will be convenient to consider the union ofḠ 1 andḠ 2 obtained by putting together all the vertices and lines ofḠ 1 andḠ 2 . This is clearly a graph which we denoteḠ 1 ∪Ḡ 2 . Its number of external legs is given by Using Eqs.
(1)-(4), it is then easily checked that This formula strongly reminds the well known relation between the cardinals of two finite sets X 1 , X 2 and the cardinals of the sets X 1 ∪ X 2 and X 1 ∩ X 2 . 4 We stress however that Eq. (5) is not a trivial application of the corresponding formula between the cardinals of the sets of external legs ofḠ 1 ,Ḡ 2 ,Ḡ 1 ∪Ḡ 2 andḠ 1 ∩Ḡ 2 because the sets of external legs ofḠ 1 orḠ 2 are not subsets of the set of external legs ofḠ 1 ∪Ḡ 2 and so the union of the sets of external legs ofḠ 1 andḠ 2 is not the set of external legs ofḠ 1 ∪Ḡ 2 . Instead, the formula (5) needs to be seen as consequence of the overlapping structure depicted in Fig. 3.

C. Listing the possible overlaps
The previous formulas allow us to list all possible overlaps betweenḠ 1 andḠ 2 . First it follows from Eq. (5) that a necessary condition forḠ 1 andḠ 2 to have an overlap is that This also means that, given n ext (Ḡ 1 ) and n ext (Ḡ 2 ), we can obtain all possible overlaps between G 1 andḠ 2 by considering all possible values of n ext (Ḡ 1 ∩Ḡ 2 ) compatible with the constraint (6) and, for each of these values, solve the system (1)-(3) for x, x 1 and x 2 as a function of n t 1 and n t 2 . One finds x 1 = n ext (Ḡ 1 ) − n ext (Ḡ 1 ∩Ḡ 2 ) + n t 1 , with the constraints Any possible solution defined by the values of x, x 1 , x 2 , n t 1 and n t 2 is called an overlap mode. In App. A, we determine the number of overlap modes and in Table I, we collect the various overlap modes for subgraphs with 0 and 1 external legs. An example of overlap of two subgraphs with 6 external legs each is provided in Fig. 1 where the highlighted subgraph, obtained by cutting the lines attached to the legs e 3 , . . . , e 6 , overlaps with the subgraph obtained by cutting the lines attached to the legs e 1 and e 2 . This overlap mode is characterized by n t 1 = 4 (with n c 1 = 2 and n r 1 = 1), n t 2 = 2 (with n c 2 = 2 and n r 2 = 0), We mention that the above equations make no direct reference to the numbers or connecting lines n c i or to the numbers of returning lines n r i , but rather to the combination n t i = n c i + 2n r i and, in fact, one can interprete the returning lines as a degenerate case of connecting line which does not connect to any complementary graph but loops back instead to the subgraph. This allows to simplify the graphical representation given in Fig. 3 by ignoring the returning lines, or, more precisely, by hiding them as part of the connecting lines. 5 overlap modes between subgraphs with 0 and 1 external legs as obtained using Eqs. (7), (8) and (9) and the constraints (10)- (12). We have omitted certain cases that are deduced from the ones in the table using the exchangeḠ 1 ↔Ḡ 2 . For the cases listed here n t i ≤ 1 and thus there are no returning lines.
So far the analysis concerned any type of subgraphs of a given graph. In particular, the subgraphs did not need to be connected. In the next section, we particularize the analysis to specific classes of subgraphs for which we show that overlaps are not possible.

IV. THE CASE OF ONE-PARTICLE-IRREDUCIBLE SUBGRAPHS
A one-particle-irreducible (1PI) subgraph is a connected graph that cannot be made into two disconnected pieces by cutting just one line. In this section we analyze the possibility of overlap between two such subgraphs. More precisely, we look for generic enough conditions under which such overlaps are excluded. Of course, it will be implicitly assumed here that none of the subgraphs in question is a subgraph of the other one (in particular, they are assumed to be distinct). Otherwise they always overlap, in a trivial manner. Moreover, since the union of two overlapping 1PI subgraphs is also 1PI, it is necessarily contained in one of the 1PI components of the original graph G. Thus, without loss of generality, we can assume that the original graph is 1PI (and in particular connected).
A 1PI subgraph with p external legs is called a p-insertion. This notion includes the (1PI) graph itself if the latter has p external legs, and also any single vertex associated to the ϕ p interaction if the latter was included in the theory. We shall now introduce certain notions associated to p-insertions and then analyze the conditions under which the 2-, 3and 4-insertions cannot overlap.

A. Definitions
A graph G is called a p-skeleton if it contains no other p-insertion than the graph itself (if it has p external legs) or those made of a single vertex (if the ϕ p interaction vertex is part of the model). We mention that a 1PI graph is necessarily a 0-skeleton (since it is connected) and also a 1-skeleton (since any non-trivial 1-insertion would be necessarily connected to the rest of the graph by a line). More generally, we call p 1 /p 2 -skeleton, a graph that is both a p 1 -and a p 2 -skeleton.
Given two p-insertionsḠ 1 andḠ 2 of a graph G, it might occur that one of them is a subgraph of the other, sayḠ 1 ⊂Ḡ 2 . This defines a partial ordering over the set of p-insertions of the graph G. As any partial ordering over a finite set, it admits maximal elements, that is elements that are larger than any other element that is ordered with respect to them.
In the present context, we refer to these maximal elements as maximal p-insertions. They correspond to p-insertions that are not themselves subgraphs of another p-insertion within the graph G. Obviously, a maximal p-insertion cannot be a subgraph of another maximal p-insertion of the same graph, unless these two maximal p-insertions coincide.
The union of two overlapping p-insertions is another q-insertion. This is because, if there was a way to split the resulting graph by cutting one line, the cut should lie in any of the two original p-insertions. But this is impossible since the latter are 1PI by definition. If we now consider the particular case of the union of two overlapping (and distinct) maximal p-insertions, then necessarily q = p since otherwise the union would have created a new p-insertion that is distinct from the original ones and that is larger than any of them, in contradiction with the fact that the latter were both assumed to be maximal. We shall make use of this result below.

B. Non-overlap theorems
It is not very difficult to see what is the added value of considering 1PI subgraphs. First, the number of connecting lines ofḠ 1 ∩Ḡ 2 withinḠ i is either n c i = 0 or n c i ≥ 2, and in the first case, we necessarily have n r i ≥ 1, otherwise one subgraph would be included in the other one. It follows that n t i ≥ 2. From Eq. (3) this implies n ext (Ḡ 1 ∩Ḡ 2 ) ≥ 4, and, combining this with Eq. (5), we arrive at For not two large values of n ext (Ḡ 1 ) and n ext (Ḡ 2 ) this is a strong constraint on n ext (Ḡ 1 ∪Ḡ 2 ) that will allow us finding certain obstructions to the presence of overlaps.
Consider first the case of 2-insertions, with p 1 ≡ n ext (Ḡ 1 ) = 2 and p 2 ≡ n ext (Ḡ 2 ) = 2. From Eq. (13) it follows that n ext (Ḡ 1 ∪Ḡ 2 ) = 0. Since the original graph G is assumed to be 1PI, this means thatḠ 1 ∪Ḡ 2 is the graph G itself, and, therefore, that the latter cannot have any external leg. We have thus arrive at a first "non-overlap" theorem: the only possibility for a overlap of 2-insertions (self-energies) is within a graph with no external legs. In other words: Theorem 2: 2-insertions cannot have an overlap within a (1PI) graph with external legs.
Let us mention that we know exactly how this overlap occurs in the case of a graph with no external legs since n ext (Ḡ 1 ∩Ḡ 2 ) = 4 from Eq. (5) and therefore n t 1 = n t 2 = 2 from Eqs. (10)- (12).
Consider next p 1 ≡ n ext (Ḡ 1 ) = 3 and p 2 ≡ n ext (Ḡ 2 ) = 3. In this case, the inequality (13) leaves room for the cases n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1, 2. We could analyze the various overlap modes using the discussion in the previous section. However, our purpose here is to find conditions for non-overlap. To this purpose, we note that in the cases n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1, we have againḠ 1 ∪Ḡ 2 =Ḡ and therefore these cases can only exist if the original (1PI) graph G has 0 or 1 external legs. In the case n ext (Ḡ 1 ∪Ḡ 2 ) = 2, we do not necessarily haveḠ 1 ∪Ḡ 2 =Ḡ, so this case is possible if the original graph has two external legs or if it contains 2-insertions.
We then arrive at a second non-overlap theorem: Theorem 3: 3-insertions cannot have a overlap within a (1PI) 2-skeleton graph that has strictly more than two external legs.
Let us finally consider p 1 ≡ n ext (Ḡ 1 ) = 4 and p 2 ≡ n ext (Ḡ 2 ) = 4. In this case, the inequality (13)  graph that has strictly more than three external legs.
The above results have been obtained by using the inequality (13). An altenerative strategy consists in listing all possible overlaps of a given type and check that none of them fulfills the premises of the above theorems. This is done in Fig. 4 where the possible overlaps between 2-, 3-and 4-insertions are listed. In each figure, the blobs make reference to the blobs in Fig. 3, with the little difference that we have hidden the returning lines as part of the connecting lines, see the discussion at the end of Sec. III C.

C. Overlapping insertions of different order
We can also consider overlaps between insertions of different order. Take for instance p 1 = 2 and p 2 = 3. The inequality (13) implies n ext (Ḡ 1 ∪Ḡ 2 ) ≤ 1. Thus an overlap between these insertions can only occur in graphs with zero or one external leg, or, in other words, a 2-insertion and a 3-insertion cannot have an overlap within a graph with strictly more than one leg: Theorem 23: A 2-insertion and a 3-insertion cannot overlap within a (1PI) graph with strictly more than one external leg.
This is a situation similar to the one we encountered for the overlap of two 4-insertions: the highest possible value of n ext (Ḡ 1 ∪Ḡ 2 ) allowing for an overlap, that is 2, coincides precisely with the number of external legs of one of the insertions we are probing. Consider then not an overlap between an arbitrary 2-insertion and an arbitrary 4-insertion, but rather between a maximal 2-insertion and an arbitrary 4-insertion. This type of overlap cannot occur in a graph with strictly more than one external leg. Indeed, the cases n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1 are trivially excluded, whereas the case n ext (Ḡ 1 ∪Ḡ 2 ) = 2 is excluded because otherwiseḠ 1 ∪Ḡ 2 would correspond to a 2-insertion that contains strictlyḠ 1 , in contradiction with the fact thatḠ 1 was assumed to be maximal. We arrive then at the following result Theorem 24: A maximal 2-insertion and a 4-insertion cannot overlap within a (1PI) graph with strictly more than one external leg.
Finally, for p 1 = 3 and p 2 = 4, an overlap can occur only in the cases n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1, 2, 3. This is a situation similar to the one we encountered for the overlap of two 4-insertions: the highest possible value of n ext (Ḡ 1 ∪Ḡ 2 ) allowing for an overlap, that is 3, coincides precisely with the number of external legs of one of the insertions we are probing.
Consider then not an overlap between an arbitrary 3-insertion and an arbitrary 4-insertion, but rather between a maximal 3-insertion and a arbitrary 4-insertion. This type of overlap cannot occur in a 2-skeleton graph with strictly more than two external legs. Indeed, the cases n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1, 2 are excluded in a trivial way, whereas the case n ext (Ḡ 1 ∪Ḡ 2 ) = 3 is excluded because otherwiseḠ 1 ∪Ḡ 2 would correspond to a 3-insertion that contains strictlȳ G 1 , in contradiction with the fact thatḠ 1 was assumed to be maximal. We arrive then at the following result

Theorem 34: A maximal 3-insertion and a 4-insertion cannot overlap within a (1PI)
2-skeleton graph with strictly more than two external legs.
These results can once again be derived by listing all possible overlaps between 2-, 3-and 4-insertions, see Fig. 5.

D. Connecting lines versus returning lines
So far, we made no distinction between connecting and returning lines. This was possible because they play essentially the same role. In particular, with insertions, we have n c i ≥ 2 or, in the case where n c i = 0, 2n r i ≥ 2 which allowed us to use n t i ≥ 2. One may want to make a distinction between connecting lines and returning lines, and, in particular treat the cases n c i = 0 and n c i ≥ 2 separately. When proceeding this way, one is lead to discuss three cases of overlap, a generic overlap with n c 1 ≥ 2 and n c 2 ≥ 2 and non-generic overlaps with n c 1 = 0 or n c 2 = 0, or both. Using a terminology that we introduced above, the generic overlap corresponds to the case whereḠ 1 ∩Ḡ 2 is neither dense withinḠ 1 nor within G 2 , whereas the non-generic overlaps correspond to the cases whereḠ 1 ∩Ḡ 2 is dense either withinḠ 1 or withinḠ 2 , or within both.
There is nothing to add to the discussion in the previous section in the case of a generic overlap. In the case of non-generic overlaps, however, the analysis can be slightly refined.
However, in the case p 1 = p 2 = 3, we find n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1 which is the stronger then the constraint that we foud earlier and which allows to enlarge the premise of theorem 3 to the case of graphs with strictly more than one external leg. Similarly, for p 1 = p 2 = 4, we find n ext (Ḡ 1 ∪Ḡ 2 ) = 0, 1, 2 which allows to enlarge the premise of theorem 4 to graphs with strictly more than two external legs and to any type of 4-insertion, not necessarily maximal. 6 In the case where p 1 = p 2 , with p i = 2, 3 or 4, it is easily checked that the constraints are the same as those obtained above so the premises of theorems 23, 24 and 34 are not changed.
We can find nonetheless a non-overlap theorem in the case p = 5 if we further restrict the possible subgraphs under consideration. Assume for instance that we inquire about the overlap of two-particle-irreducible (2PI) subgraphs, that is subgraphs that cannot be split apart by cutting two lines. In the case of a generic overlap, we have n t i ≥ 3 and therefore n ext (Ḡ 1 ∩Ḡ 2 ) ≥ 6, from which it follows now that This For simplicity, we first show how this is done for the vertex functions Γ (n) with n ≥ 5. To do so, we show that these vertex functions admit a skeleton expansion, that is rather than computing them by adding all the perturbative graphs they are made of, we can alternatively sum over all the 2/3/4-skeleton graphs in this list, and then replace each free propagator by the full two-point function G = [Γ (2) ] −1 , each tree-level 3-vertex by the full three point function Γ (3) and each tree-level 4-vertex by the full four-point function Γ (4) . As already mentioned in the Introduction, this is a known result. It is however interesting to see how it derives from the non-overlap theorems of the previous section. At the end of the section, we argue that this result extends in fact to a larger class of functions.

A. Hiding the bare mass
Consider a 1PI graph G with external legs. We define a chain of G as any connected sequence of lines G 0 and 2-insertions Σ i of the form Since we have assumed that the 1PI graph G has external legs, the starting G 0 is necessarily different from the ending one. Moreover, we request that this sequence is complete, that is that one cannot add additional Σ k G 0 's or G 0 Σ k 's. In a graph with external legs, it is always possible to identify unambiguously all the chains. It may happen that certain lines G 0 are not connected to any self-energy. We call these trivial chains.
Given two chains C 1 and C 2 of G, we say that C 2 is a subchain of C 1 if it is a chain of one of the 2-insertions of C 1 . This relation which we denote as C 2 ⊂ C 1 defines a partial ordering over the set of chains of G. As any partial ordering over a finite set, it admits maximal elements which we call maximal chains. Now, according to theorem 2 above, in a graph with external legs, 2-insertions cannot have any overlap (unless of course one of them is a subgraph of the other). It is then easily verified that maximal chains cannot have any overlap either (unless of course one of them is a subchain of the other). Let us now use this result to show how to hide the bare mass in Γ (n) , with n ≥ 3.
We start by writing Γ (n≥3) as where the sum runs over all Feynman graphs D that contribute to Γ (n≥3) . Depending on the context, D denotes the graph itself or the corresponding Feynman integral. It depends on the bare mass m b via the bare free propagator G 0 . We have also made explicit the dependence on the various bare couplings {g Since maximal chains do not overlap in the graphs contributing to Γ (n≥3) , one can unambiguously associate to each graph D, a 2-skeleton graph denoted D 2 and obtained from D by replacing any maximal chain by a trivial chain. It is convenient to momentarily associate a different label to each trivial chain appearing in D 2 , so that D 2 is a function of these various The original graph D can now be written in terms of its associated 2-skeleton as where the C i are the maximal chains of D that were replaced by trivial chains G i in order We mention that there is no pre-factor in the right-hand side of Eq. (17). This is because the symmetry factors factorize: the symmetry factor of D equals the symmetry factor of D 2 times the symmetry factors of the C i . This property relates to the fact that the replacement of maximal chains by trivial chains is unambiguous, see below for more details.
Let now sum both sides of Eq. (17) over the graphs D contributing to Γ (n≥3) . We perform the sum in two steps. First we sum over all graphs D that have the same D 2 , and then we sum over all the possible 2-skeletons D 2 . Because we are summing over all possible graphs of Γ (n≥3) and because this does not put any restrictions on the chains that can appear in the right-hand side of Eq. (17) for a given D 2 , we find that the sum over all graphs D that share the same D 2 replaces each chain in the right-hand side of Eq. (17) by the sum of all possible chains, that is the two-point function [Γ (2) ] −1 : which we denote for simplicity as } . We now need to sum over all possible skeletons D 2 , and we then find B. Hiding the trilinear and quartic bare couplings Let us now consider n ≥ 4 and start from Eq. (19). Since this sum is made of 2-skeletons that have strictly more than 2 legs, we can apply theorem 3. This means that to any graph D 2 , we can unambiguously associate a 2/3-skeleton D 23 ∈ P 23 by shrinking any maximal 3-insertion to a trivial one. Using the same argument as above, we find 8 Finally, let us now consider n ≥ 5 and start from Eq. (20). Since this sum is made of 2/3-skeletons that have strictly more than 3 legs, we can apply theorem 4. Using the same strategy as above, we conclude that where the sum runs over the 2/3/4-skeleton graphs contributing to Γ (n≥5) . Note that it was important to first resum the three-point function. Otherwise, the graphs would not have p , the graph rewrites in terms of the associated 2/3-skeleton as where the pre-factor α accounts for a potential mismatch between the symmetry factor of D 2 and the symmetry factor of D 23 multiplied by the symmetry factors of the V i . We now show that α = 1, meaning that the symmetry factors factorize.
To see this, we note that in order to compute the symmetry factor of } , we can first compute the symmetry factor of a (n + 3p)-point function n the number of vertices in the R, this produces a factor (n + n 1 + · · · + n p )!/(n!n 1 ! · · · n p !) in the counting of Wick contractions. If we denote by N X the number of Wick contractions of a given contribution X, we have then N D 2 = (n + n 1 + · · · + n p )! n!n 1 ! · · · n p !
But the symmetry factor is equal to the number of Wick contractions divided the factorial of the number of vertices and 3! (since we are here considering cubic vertices) elevated to the number of vertices. It follows that (24) Applying the same formula to D 23 , with V b , we find s R = s D 23 and thus which is the announced factorization of symmetry factors. The same reasoning applies to the resummation of chains or four-point functions.

D. Extension
So far we have we have considered the case of 1PI graphs. However, it is pretty clear that our results apply to a larger class of graphs. Consider first the non-overlap theorems.
They apply to any disconnected graph whose connected pieces fulfill the premises of these theorems. For instance, theorem 4 applies to any disconnected graph whose connected parts are 2/3-skeletons with strictly more than three external legs, and so on.
Next, let us wonder how the possibility to hide bare parameters extends to functions other than the Γ (n) 's. Consider for instance a quantity given as an infinite sum of 2-skeleton graphs whose connected pieces have strictly more than two external legs. It is clear that theorems 2, 3 and 23 apply to each of these graphs and one can therefore associate unambiguously 2/3-skeleton graphs to each of these graphs. If we now assume that the infinite sum of graphs puts no restriction on the 2-and 3-insertions that can appear (this is a property that needs to be verified for each infinite class of graphs that one may consider; it is of course obvious for the Γ (n) 's) then we can proceed as for the Γ (n) 's and hide the dependence on the bare mass and trilinear bare coupling using the full two-and three-point functions.
A direct application of this result is the elimination of the bare parameters in the higher derivatives δ n Φ/δG n (with n ≥ 3) of the Luttinger-Ward functional Φ[G]. This functional is the sum of two-particle-irreducible graphs with no external legs, that is graphs that cannot be split apart by cutting two lines. The derivatives δ n Φ/δG n are also sums of two-particleirreducible graphs but only with respect to cuts that leave the external legs associated to a given δ/δG on the same side of the cut. It is easily seen that the connected components of any δ n Φ/δG n with n ≥ 3 obey the premises of theorem 3 above. Moreover, the twoparticule irreductibility puts no constraint on the possible 3-insertions that can occur. 9 One can then follow the same strategy as in Sec. V B to show that δ n Φ/δG n with (n ≥ 3) admits a skeleton expansion in terms of the full two-and three-point functions. The corresponding 2/3-skeletons obey the premises of theorem 4 and since the two-particule irreductibility puts no constraint on the possible 4-insertions that can occur, one can proceed one step further and derive a skeleton expansion in terms of the full two-, three-and four-point functions. As already mentioned in the Introduction, this has been recently put into good use to formulate a finite set of flow equations for Φ-derivable approximations that make no reference to the bare theory, see Ref. [5].

A. Applications
The possibility to express Γ (n≥5) as an infinite sum of 2/3/4-skeleton graphs with propagators, three-and four-vertices given respectively by [Γ Moreover, there are no global divergences since n ≥ 5. In the case of a theory such as ϕ 6 in d = 3 dimensions, which is also renormalizable but features primarly divergent functions with 6 legs, this graphical explanation does not apply since maximal 6-point functions can overlap within any graph and therefore there are inevitably overlapping divergences.

B. Connection to nPI effective actions
The present approach pretty much resembles that followed with n-particle-irreducible (nPI) effective actions [14][15][16]. Let us here emphasize some differences however. In fact, the present approach deals only with quantities for which the bare mass and the trilinear and quartic bare coupling can be hidden into the two-, three-and four-point functions while avoiding graph over-counting. In contrast, the nPI framework deals with the sum of vacuum graphs ln Z for which none of the above non-overlap theorems apply. Indeed, for such graphs, there is no unambiguous way to identify the maximal 2-, 3-and 4-insertions. It is still possible to rewrite this sum of vacuum graphs as a sum of skeletons. 10 However, this writing always involve certain terms that depend on the bare mass and the bare couplings and requires additional terms to avoid double counting. For instance, within the 2PI framework, using the notion of cycles [14], one can show that where Φ[G] is the Luttinger-Ward function referred to above. The first two terms account for the overcounting of graphs that arise from the fact that there is no unique way to identify maximal 2-insertions in ln Z. Moreover the second term depends explicitely on the bare mass m 2 b , so it is not possible to fully hide this bare parameter in this case (although we stress that this remaining dependence is rather trivial). Similar remarks apply to the rewriting of ln Z in terms of the three-and four-point functions leading to the so-called 3PI and 4PI

VII. CONCLUSION
In this article, we have studied how two arbitrary subgraphs of a given Feynman graph can overlap with each other. When restricting to 1PI subgraphs, we have shown how this allows to derive useful "non-overlap" theorems for the cases of 2-, 3-and 4-insertions. One consequence of these is the well known skeleton expansion for vertex functions Γ (n) with n ≥ 5 which allows one to entirely hide any reference to the bare mass, as well as the trilinear and quartic bare couplings using the two-, three-and four-point functions, and this without any over-counting correction. We have also discussed how this result can be extended to other classes of functions, in particular to iterated derivatives of the Luttinger-Ward functional.
As discussed in Ref. [5], the previous results have applications in the renormalization of the 2PI effective action and the corresponding Φ-derivable approximations, as well as in the construction of new truncation schemes for the functional renormalization group hierarchy but their potential range of applicability is definitely larger. In this section, we would like to evaluate the number of overlap modes of two graphsḠ 1 andḠ 2 of n ext (Ḡ 1 ) and n ext (Ḡ 2 ) external lines each, when no restrictions are imposed on the graphs as in Sec. III. These are determined by the constraints (6) and (10)- (12). To ease the reading we shall simplify the notations as n 1 ≡ n ext (Ḡ 1 ), n 2 ≡ n ext (Ḡ 2 ) and n ≡ n ext (Ḡ 1 ∩Ḡ 2 ).
Without loss of generality, we can assume that n 1 ≤ n 2 .
We first need to consider all possible values of n compatible with (A1), that is n 1 + n 2 + 1 choices in total. For each of these values, we need to choose n t 1 and n t 2 complying with (A2)-(A4). Here, we need to consider various ranges for the values of n over which (A3) and/or (A4) do not matter. More precisely, in the range n ≤ n 1 , none of these constraints matter, in the range n 1 < n ≤ n 2 only one of them matters, and in the range n 2 < n ≤ n 1 + n 2 , both of them. We will use the following result: given three positive integers a, b, c such that a ≥ b ≥ c and c ≤ a − b, the conditions n 1 + n 2 ≤ a, n 1 ≥ b, n 2 ≥ c define an isosceles right triangle in the (n 1 , n 2 )-plane of sides of length a − b − c, corresponding to (a − b − c + 1)(a − b − c + 2)/2 points.
The formula for arbitrary n 1 and n 2 (that is not necessarily ordered as n 1 ≤ n 2 ) is obtained after replacing n 1 by Min(n 1 , n 2 ) and n 2 by Max (n 1 , n 2 ). For n 1 = n 2 = 0, we find 1. For n 1 = 0 and n 2 = 1, we find 2. For n 1 = n 2 = 1, we find 5. This agrees with the number of overlap modes found in Tab. I. [2] G. 't Hooft, arXiv:hep-th/0405032 [hep-th].