Graphs and Combinatorics (2009) 25:717–726 DOI 10.1007/s00373-010-0878-0 ORIGINAL PAPER Elementary Graphs with Respect to f -Parity Factors Mikio Kano · Gyula Y. Katona · Jácint Szabó Received: 30 September 2008 / Revised: 16 July 2009 / Published online: 6 February 2010 © Springer 2010 Abstract This note concerns the f -parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with deg F ≤ f and minimizing the number of vertices x with deg F (x) ≡ f (x) mod 2. First we prove a Gallai–Edmonds type structure theorem and some other known results on the f -parity subgraph problem, using an easy reduction to the matching problem. Then we use this reduction to investigate barriers and elementary graphs with respect to f -parity factors, where an elementary graph is a graph such that the union of f -parity factors form a connected spanning subgraph. Keywords f -Parity factor · Parity factor · (1, f )-Odd factor · Odd factor · Elementary graph · Structure theorem M. Kano’s research is supported by Grant-in-Aid for Scientific Research of Japan, G.Y. Katona’s research is partially supported by the Hungarian National Research Fund and by the National Office for Research and Technology (Grant Number OTKA 67651 and 78439), J. Szabó’s research is supported by OTKA grants T037547, K60802, TS 049788 and by European MCRTN Adonet, Contract Grant No. 504438. M. Kano (B) Department of Computer and Information Sciences, Ibaraki University, Hitachi, Ibaraki 316-8511, Japan e-mail: kano@mx.ibaraki.ac.jp G. Y. Katona Department of Computer and Information Sciences, Technical University of Budapest, P.O. Box 91, Budapest 1521, Hungary e-mail: kiskat@cs.bme.hu J. Szabó Egerváry Research Group (EGRES), Department of Operations Research, Eötvös University, Pázmány P. s. 1/C, Budapest 1117, Hungary 123 718 Graphs and Combinatorics (2009) 25:717–726 1 Introduction In this paper we deal with a special case of the degree prescribed subgraph problem, introduced by Lovász [10]. This is as follows. Let G be an undirected graph and let ∅ = Hv ⊆ N ∪ {0} be a degree prescription for each v ∈ V (G). For a spanning F (v) = min{| deg (v) − i| : i ∈ H }, and let subgraph F of G, define δH v F F δH =
v∈V (G) F F δH (v) and δH (G) = min δH , F where the minimum is taken over all the spanning subgraphs F of G. A spanning F = δ (G), and it is an H-factor if δ F = 0, subgraph F is called H-optimal if δH H H i.e., if deg F (v) ∈ Hv for all v ∈ V (G). The degree prescribed subgraph problem is to determine the value of δH (G). An integer h is called a gap of H ⊆ N ∪ {0} if h ∈ / H but H contains an element less than h and an element greater than h. Lovász [12] gave a structural description on the degree prescribed subgraph problem in the case where Hv has no two consecutive gaps for all v ∈ V (G). He showed that the problem is NP-complete without this restriction. The first polynomial time algorithm was given by Cornuéjols [2]. It is implicit in Cornuéjols [2] that this algorithm implies a Gallai–Edmonds type structure theorem for the degree prescribed subgraph problem (first stated in [14]), which is similar to—but in some respects much more compact than—that of Lovász’. The case when an odd value function f : V (G) → N is given and Hv = {1, 3, 5, . . . , f (v)} for all v ∈ V (G), is called the (1, f )-odd subgraph problem. We denote δH (G) = δ f (G). This problem is much simpler than the general case due to the fact that only parity requirements are posed. The (1, f )-odd subgraph problem was first investigated by Amahashi [1], who gave a Tutte type characterization of graphs having a [1, n]-odd factor, where n ≥ 1 is an odd integer. A Tutte type theorem for a general odd value function f was proved by Cui and Kano [3], and then a Berge type minimax formula on δ f (G) by Kano and Katona [7]. A Gallai–Edmonds type theorem on the (1, f )-odd subgraph problem was given in [8] and [14]. In this note we show a new approach to the (1, f )-odd subgraph problem. Actually, it is worth allowing f to have also even values and defining Hv equal to {1, 3, . . . , f (v)} or {0, 2, . . . , f (v)}, according to the parity of f (v). We call this the f -parity subgraph problem. We show an easy reduction of the f -parity subgraph problem to the matching problem, and we show that this reduction easily yields the above mentioned Gallai–Edmonds and Berge type theorems on the f -parity subgraph problem. Then we investigate barriers with respect to the f -parity subgraph problem. As another application, we explore the graphs for which the edges belonging to some f -parity factor form a connected spanning subgraph. We call such a graph an f -elementary graph. We generalize some results on matching elementary graphs (proved by Lovász [11]) to f -elementary graphs. An attempt putting the f -parity subgraph problem into the general context of graph packing problems can be found in [15]. The f -parity subgraph problem can be reduced to the (1, f )-odd subgraph problem by the following construction: for every vertex v ∈ V (G) with f (v) even, connect 123 Graphs and Combinatorics (2009) 25:717–726 719 a new vertex wv to v in G, define f (wv ) = 1 and increase f (v) by 1. Now δ f (G) remains the same. To avoid minor technical difficulties we assume that f > 0. Almost all results of the paper would hold without this restriction, too. Note that if G is a nontrivial f -elementary graph then f > 0 always holds. The constant function f ≡ 1 is simply denoted by 1. For X ⊆ V (G), let G[X ] be the subgraph  of G induced by X , Ŵ(X ) = {y ∈ V (G) − X : ∃ x ∈ X, x y ∈ E(G)}, f (X ) = { f (x) : x ∈ X } and χ X denote the function with χ X (x) = 1 if x ∈ X and χ X (x) = 0 otherwise. The number of components of a graph G is denoted by ω(G). The cardinality of a set X is written |X |, and N denotes the set of natural numbers. Two isomorphic graphs H and K are denote by H ≃ K . The graphs are finite and undirected, and have no loops, but may have multiple edges. 2 Reduction to Matchings In this section we show a reduction of the f -parity subgraph problem to matchings, which will then be used to prove the Gallai–Edmonds type structure theorem on the f -parity subgraph problem. The auxiliary graph we use is defined below. Definition For a graph G and a function f : V (G) → N, define G f to be the following undirected graph. Replace every vertex v ∈ V (G) by a new complete graph on f (v) vertices, denoted by K f (v) , and for each pair of vertices u, v ∈ V (G) adjacent in G, add all possible f (u) f (v) edges between K f (u) and K f (v) . Let V f (v) = V (K f (v) ). Observe that G 1 = G, |V (G f )| = f (V (G)) and that V f (v) = ∅ for every v ∈ V (G). There is a strong connection between the maximum matchings of G f and the optimal f -parity subgraphs of G. Note that the size of a maximum matching of G is just |V (G)| − δ1 (G). Lemma 1 For every optimal f -parity subgraph F of G, there exists a matching M of G f such that |V (M)| = f (V (G)) − δ Ff . Moreover, if deg F (w) ∈ {. . . , f (w) − 3, f (w) − 1} for a vertex w ∈ V (G) then M can be chosen to miss a prescribed vertex x ∈ V f (w) . On the other hand, for every maximum matching M of G f there exists a spanning subgraph F of G such that δ Ff = f (V (G)) − |V (M)|. Moreover, if M misses a vertex in V f (w) for some w ∈ V (G) then F can be chosen such that deg F (w) ∈ {. . . , f (w) − 3, f (w) − 1}. In particular, δ f (G) = δ1 (G f ). Proof Let F be an optimal f -parity subgraph of G. Assume deg F (w) ∈ {. . . , ′ f (w) − 3, f (w) − 1}. If deg F (u) > f (u) for some u ∈ V (G) then clearly δ Ff ≤ δ Ff for the spanning subgraph F ′ that is obtained from F by deleting an edge e incident to u. As F is f -parity optimal, e is not adjacent to w, so deg F ′ (w) = deg F (w). Hence we assume that deg F (v) ≤ f (v) for every vertex v. A matching M of G f is obtained from F as follows. For every edge x y of F, M contains exactly one edge joining K f (x) to K f (y) . Then for every K f (v) , if deg F (v) ≡ f (v), then M contains edges that covers all the remaining f (v) − deg F (v) vertices of K f (v) , and otherwise M contains edges covering all the remaining vertices but one vertex. Therefore a matching M of G f 123 720 Graphs and Combinatorics (2009) 25:717–726 misses exactly δ Ff vertices. For a vertex w with deg F (w) ∈ {. . . , f (w)−3, f (w)−1}, M can be chosen to miss a prescribed vertex in V f (w) . For the second part, let M be a maximum matching of G f . If M contains two edges between K f (x) and K f (y) for some x, y ∈ V (G), then replace them by two edges, one inside K f (x) and the other one inside K f (y) . Thus we may assume that M contains at most one edge between K f (u) and K f (v) for all distinct u, v ∈ V (G). By contracting each K f (u) to one vertex u, we get a spanning subgraph F of G with δ Ff = f (V (G)) − |V (M)|. Moreover, deg F (w) ∈ {. . . , f (w) − 3, f (w) − 1} in the case that M misses a vertex in K f (w) . We define critical graphs with respect to the f -parity subgraph problem as in the matching case. If f = 1 the graphs defined below are called factor-critical. Definition Given a graph G and a function f : V (G) → N, G is called f -critical if for every w ∈ V (G) there exists a spanning subgraph F of G such that deg F (w) ∈ {. . . , f (w)−3, f (w)−1} and deg F (v) ∈ {. . . , f (v)−2, f (v)} for all v ∈ V (G)−{w}. By Lemma 1, G is f -critical if and only if G f is factor-critical. The Gallai–Edmonds structure theorem for the f -parity subgraph problem follows from the classical Gallai–Edmonds theorem easily. We cite this latter result below. Theorem 2 [4–6] Let D consist of those vertices of a graph G which are missed by some maximum matching of G, let A = Ŵ(D) and C = V (G) − (D ∪ A). Then 1. 2. 3. 4. every component of G[D] is factor-critical, |{K : K is a component of G[D] adjacent to A′ }| ≥ |A′ |+1 for all ∅ = A′ ⊆ A, δ1 (G) = ω(G[D]) − |A|, G[C] has a perfect matching. A direct generalization of the above result is the version for the f -parity subgraph problem. Theorem 3 [8,14] Let G be a graph and f : V (G) → N be a function. Let D f ⊆ V (G) consist of those vertices v for which there exists an optimal f -parity subgraph F of G with deg F (v) ∈ {. . . , f (v) − 3, f (v) − 1}. Let A f = Ŵ(D f ) and C f = V (G) − (D f ∪ A f ). Then 1. every component of G[D f ] is f -critical, 2. |{K : K is a component of G[D f ] adjacent to A′ }| ≥ f (A′ )+1 for all ∅ = A′ ⊆ Af, 3. δ f (G) = ω(G[D f ]) − f (A f ), 4. G[C f ] has an f -parity factor. The above Theorem 3 follows immediately from the next Lemma 4 and Lemma 1. Lemma 4 Consider the Gallai–Edmonds decomposition D ∪ A ∪ C of G f and the decomposition D f ∪ A f ∪ C f of G given in Theorem 3. Then for X = D, A, C, it holds that X f = {v ∈ V (G) : V f (v) ⊆ X }. 123 Graphs and Combinatorics (2009) 25:717–726 721 Proof Assume V f (v) meets D. Then there exists a maximum matching M of G f that misses exactly one vertex of V f (v) . By Lemma 1, for any vertex z of V f (v) , there exists a maximum matching of G f that misses z, and hence V f (v) ⊆ D. Thus D f = {v ∈ V (G) : V f (v) ⊆ D} by Lemma 1. By the construction of G f , if V f (v) meets A, then V f (v) ⊆ A, and so A f = {v ∈ V (G) : V f (v) ⊆ A}. Therefore the lemma also holds for X = C. From Theorem 3 the Berge type minimax formula on the f -parity subgraph problem follows easily. Definition A component K of G is called f -odd or f -even when f (V (K )) is odd or even. Let f -odd(G) denote the number of f -odd components of G. Let def f (Y ) = f - odd(G − Y ) − f (Y ) for Y ⊆ V (G). Theorem 5 [7] For a graph G and a function f : V (G) → N, it follows that δ f (G) = max{def f (Y ) : Y ⊆ V (G)}. Proof Let Y ⊆ V (G). Since an f -odd component K of G − Y has no f -parity factor, it follows that δ Ff ≥ f -odd(G − Y ) − f (Y ) = def f (Y ) for every spanning subgraph F of G, and thus δ f (G) ≥ def f (Y ). By virtue of Theorem 3 and by the fact that every f -critical component of G − A f is f -odd, we have δ f (G) = ω(G[D f ]) − f (A f ) = f -odd(G − A f ) − f (A f ) = def f (A f ). Hence the theorem holds. Now we show how to use this approach to analyze barriers. Definition A set Y ⊆ V (G) is called an f -barrier if def f (Y ) = δ f (G). As f -critical graphs are f -odd, the canonical Gallai–Edmonds set A f is an f -barrier. A 1-barrier is just an ordinary barrier in matching theory. One can observe that if Y ⊆ V (G f ) satisfies V f (v) ∩ Y = ∅ and V f (v) \ Y = ∅, then V f (v) ∩ Y is adjacent to only one component of G f − Y . Moreover, if Y is a barrier in G f then each X ⊆ Y is adjacent to at least |X | odd components of G f − Y since otherwise def 1 (Y − X ) = f -odd(G − (Y − X )) − |Y − X | > f -odd(G − Y ) − |X | − (|Y | − |X |) = def 1 (Y ), which is impossible. Hence if Y is a barrier in G f then |Y ∩ V f (v) | ∈ {0, 1, f (v)} for all v ∈ V (G). It also follows that if |Y ∩ V f (v) | = 1 and V f (v) \ Y = ∅ then Y \ V f (v) is a barrier of G f since the unique vertex in Y ∩ V f (v) is adjacent to exactly one odd component of G f − Y containing V f (v) \ Y . Thus if Y is a barrier of G f then Y ′ = {v ∈ V (G) : V f (v) ⊆ Y } is an f -barrier of G. On the other hand, if Y ′ is an f -barrier of G then {V f (v) : v ∈ Y ′ } is clearly a barrier of G f . The canonical Gallai–Edmonds barrier A(G f ) of G f has this form. 123 722 Graphs and Combinatorics (2009) 25:717–726 Definition An f -barrier Y of G is called strong if the f -odd components of G − Y are f -critical. It is obvious that A f is a strong f -barrier. Since a graph K is f -critical if and only if K f is factor-critical, we have the following.  Observation A set Y ⊆ V (G) is a strong f -barrier in G if and only if {V f (v) : v ∈ Y } is a strong 1-barrier in G f . Király proved that the intersection of strong 1-barriers is also a strong 1-barrier [9]. This result holds for the f -parity subgraph problem as well. Theorem 6 The intersection of strong f -barriers is a strong f -barrier.  Proof Let Y1 , Y2 be strong f -barriers of G. Then Yi′ = {V f (v) : v ∈ Yi } are strong  1-barriers of G f , hence their intersection, which is just {V f (v) : v ∈ Y1 ∩ Y2 }, is also a strong 1-barrier by [9]. Thus Y1 ∩ Y2 is a strong f -barrier of G. By Tutte’s theorem, maximal barriers for matching are strong. This remains true for f -barriers, too. Indeed, let Y be a maximal f -barrier of G and K be an f -odd component of G − Y . Then K has no f -parity factor, and so C f (K ) = V (K ) in its canonical Gallai–Edmonds decomposition. Hence either D f (K ) = V (K ) or A f (K ) = ∅. In the first case K is f -critical by Theorem 3, and in the second case Y ∪ A f (K ) would be a larger f -barrier than Y , which is impossible. Thus all f -odd components of G − Y are f -critical, implying that Y is strong. In the matching case, it holds that the canonical Gallai–Edmonds barrier A is the intersection of all maximal barriers. This fails for the general case: take a triangle together with a pendant vertex w of degree 1, and define f ≡ deg. Then this graph is of order four and has an f -parity factor, which is a whole graph, and A f = ∅. But it has exactly one nonempty barrier {w}. However, the fact that in matchings the canonical Gallai–Edmonds barrier A is the intersection of all strong barriers remains true for f -parity subgraphs by the above observation and the fact that A f itself is strong. 3 f -Elementary Graphs In this section we generalize some results on elementary graphs, obtained in Lovász [11], to the f -parity case. Definition Let G be a connected graph and f : V (G) → N. An edge e ∈ E(G) is said to be f -allowed if G has an f -parity factor containing e. Otherwise e is f -forbidden. The graph G is said to be f -elementary if the f -allowed edges induce a connected spanning subgraph of G. The graph G is weakly f -elementary if G 2 is f -elementary, where G 2 is the graph obtained from G by replacing every edge e ∈ E(G) by two parallel edges. A 1-elementary graph is briefly called elementary. An f -elementary graph is weakly f -elementary, but not vice versa: G = K 2 with f ≡ 2 is weakly f -elementary but not f -elementary. These classes coincide if f = 1. Lemma 7 justifies why we introduced the weak version of f -elementary graphs. 123 Graphs and Combinatorics (2009) 25:717–726 723 Lemma 7 G f is elementary if and only if G is weakly f -elementary. Proof Let M be a perfect matching of G f . If M contains at least three edges between K f (u) and K f (v) for some u, v ∈ V (G), then replace two of them by another two edges, one inside K f (u) and the other one inside K f (v) . So the number of edges of M between K f (u) and K f (v) decreased by 2. This construction shows that if G f is elementary then G is weakly f -elementary. On the other hand, if G is weakly f -elementary then G f is clearly elementary. The f = 1 special cases of the following two theorems can be found in Lovász and Plummer [13] (Theorems 5.1.3 and 5.1.6). Using our reduction, these special cases together with Lemmas 4 and 7 imply both Theorem 8 and 9. Theorem 8 A graph G is weakly f -elementary if and only if δ f (G) = 0 and C f −χw (G) = ∅ for all w ∈ V (G). Proof A graph G is weakly f -elementary if and only if G f is elementary by Lemma 7, and G f is elementary if and only if δ1 (G f ) = 0 and C(G f − x) = ∅ for all x ∈ V (G f ) [13, Theorem 5.1.3]. Since δ f (G) = δ1 (G f ), it is enough to prove that under the assumption δ f (G) = δ1 (G f ) = 0, for every w ∈ V (G) it follows that C(G f − x) = ∅ for every x ∈ V f (w) ⇐⇒ C f −χw (G) = ∅. (1) If f (w) ≥ 2, then G f − x ≃ G f −χw and so (1) follows from Lemma 4. Thus assume that f (w) = 1. As G f − x ≃ (G − w) f , Lemma 4 implies that C(G f − x) = ∅ ⇔ C f (G − w) = ∅. Hence it suffices to show that C f (G − w) = ∅ ⇐⇒ C f −χw (G) = ∅. (2) Since δ f (G) = 0 and f (w) = 1, it is easy to see that an optimal ( f − χw )-parity subgraph of G is either an f -parity factor of G or an optimal f -parity subgraph of G − w enlarged by w as an isolated vertex, and vice versa. Since ( f − χw )(w) = 0, we have w ∈ D f −χw (G) by the definition of D f −χw (G). Thus D f −χw (G) = D f (G − w). For the edge e = wu of an f -parity factor F of G, F − e is an optimal ( f − χw )-parity subgraph of G, and hence u ∈ D f −χw (G) and w ∈ A f −χw (G). It is immediate that A f −χw (G) − {w} = A f (G − w). Therefore (2) holds. Theorem 9 A graph G is weakly f -elementary if and only if f -odd(G − Y ) ≤ f (Y ) for all Y ⊆ V (G), and if equality holds for some Y = ∅ then G − Y has no f -even components. Proof Call Y ⊆ V (G) f -bad if either f -odd(G − Y ) > f (Y ) or equality holds here and G − Y has an f -even component. It follows from Lemma 7 that the graph G is weakly f -elementary if and only if G f is elementary, which is equivalent to that G f an f -bad has no 1-bad set [13, Theorem 5.1.6]. So we only have to prove that G has set Y if and only if G f has a 1-bad set Y ′ . If Y ⊆ V (G) is f -bad then Y ′ = {V f (v) : v ∈ Y } is 1-bad in G f . On the other hand, assume that Y ′ ⊆ V (G f ) is 1-bad in G f . 123 724 Graphs and Combinatorics (2009) 25:717–726 If V f (v) ∩ Y ′ = ∅ and V f (v) \ Y ′ = ∅ for some v ∈ V (G) then let x ∈ Vv ∩ Y ′ . Now x is adjacent to only one component of G f − Y ′ hence Y ′ − x is also 1-bad. So we can assume that Y ′ is a union of some V f (v) . Now Y = {v ∈ V (G) : V f (v) ⊆ Y ′ } is f -bad in G. In the matching case the existence of a certain canonical partition of the vertex set was revealed by Lovász [11] [13, Theorem 5.2.2]. We cite this result. Definition A set X ⊆ V (G) is called nearly f -extreme if δ f −χ X (G) = δ f (G) + |X |. Besides, X is f -extreme if δ f (G − X ) = δ f (G) + f (X ). It is clear that δ f −χ X (G) ≤ δ f (G) + |X | and δ f (G − X ) ≤ δ f (G) + f (X ) for every X ⊆ V (G). Nearly 1-extreme and 1-extreme sets coincide. Theorem 10 [11] If G is elementary then the maximal barriers of G form a partition S of V (G). Moreover, it holds that 1. for u, v ∈ V (G), the graph G − u − v has a perfect matching if and only if u and v are contained in different classes of S, (hence an edge x y of G is 1-allowed in G if and only if x and y are contained in different classes of S), 2. S ⊆ V (G) is a class of S if and only if G − S has |S| components, each factorcritical, 3. X ⊆ V (G) is 1-extreme if and only if X ⊆ S for some S ∈ S. Lemma 7 implies the analogue of this result. Theorem 11 If G is weakly f -elementary then its maximal f -barriers form a subpartition S ′ of V (G). Call the classes of S ′ proper, and add all elements v ∈ V (G) not in a class of S ′ as a singleton class yielding the partition S of V (G). Now it holds that 1. for u, v ∈ V (G), the graph G has an ( f − χ{u,v} )-parity factor if and only if u and v are contained in different classes of S (hence an edge x y of G is f -allowed in G 2 if and only if x and y are contained in different classes of S), 2. S ⊆ V (G) is a class of S ′ if and only if G − S has f (S) components, each f -critical, 3. X ⊆ V (G) is nearly f -extreme ( f -extreme, resp.) if and only if X ⊆ S for some S ∈ S (S ∈ S ′ , resp.). Proof Suppose that G is weakly f -elementary. Then G f is elementary. As we already observed, every barrier Y of G f satisfies |Y ∩ V f (v) | ∈ {0, 1, f (v)} for all v ∈ V (G). Since G f is elementary, its maximal barriers form a partition S f of V (G f ) by Theorem 10. Thus if a maximal barrier of G f contains exactly one vertex x of V f (u) and whole V f (v) , then by symmetry, another maximal barrier contains one vertex y ∈ V f (u) − {x} and V f (v) , which contradicts the above fact that the maximal barriers of G f is either a union of some form a partition V (G f ). Hence every maximal barrier  ′ V f (v) or a singleton. If Y is an f -barrier of G then {V f (v) : v ∈ Y ′ } is a barrier of G f . On the other hand, if Y is a maximal barrier of G f of the form V f (v) then Y ′ = {v ∈ V (G) : V f (v) ⊆ Y } is clearly a maximal f -barrier of G. So these barriers 123 Graphs and Combinatorics (2009) 25:717–726 725 Y ′ form the proper classes of S, and for a singleton class {v} ∈ S − S ′ , it holds that each vertex x ∈ V f (v) is a maximal barrier of G f . Now statement (i) is immediate from Theorem 10 (i), and (ii) also follows from Theorem 10 (ii) since if S ∈ S and |S| ≥ 2 then {Vv : v ∈ S} ∈ S f , and by the fact that a graph K is f -critical if and only if K f is factor-critical. Finally, (iii) follows from Theorem 10 and from the observation that X ⊆ V (G) is f -extreme if and only if G f has an extreme set X ′ consisting of one vertex from each Vv , v ∈ X . Remark It follows from Theorem 11 that S could be introduced as the partition {X ⊆ V (G) : X is a maximal nearly f -extreme set of G}. Besides, if X ⊆ V (G), |X | ≥ 2 is maximal nearly f -extreme, then X is an f -barrier of G. Corollary 12 If G is f -elementary then an edge e is f -allowed if and only if e joins two classes of S. Proof Suppose that e joins u to v and let g = f − χ{u,v} . By Theorem 11(i), we only have to prove that G has a g-parity factor if and only if e is f -allowed. Assume that G has a g-parity factor but e is not f -allowed (the other direction is trivial). If G − e had a g-parity factor F then F + e would be an f -parity factor of G, which is impossible. Thus by Theorem 5 there exists a set Y ⊆ V (G) such that g-odd(G − e − Y ) > g(Y ). Since G has a g-parity factor, it follows from parity reasons that g-odd(G − e − Y ) = g(Y ) + 2, and e joins two g-odd components Q 1 and Q 2 of G − e − Y . But then clearly no edge joining Y to V (Q 1 ) ∪ V (Q 2 ) is f -allowed in G. Since G is f -elementary, we have V (G) = V (Q 1 ) ∪ V (Q 2 ) and Y = ∅, but then e is an f -forbidden cut edge, which contradicts that G is f -elementary. Our last subject is generalizing bicritical graphs. Definition Let G be a graph and f : V (G) → N be a function. Then G is said to be f -bicritical if G has an ( f − χ{u,v} )-parity factor for all pairs u, v ∈ V (G). Theorem 13 If G is weakly f -elementary then the following statements are equivalent. 1. G is f -bicritical. 2. All classes of S are singletons. 3. If Y ⊆ V (G) and |Y | ≥ 2 then f -odd(G − Y ) ≤ f (Y ) − 2. Proof (i) ⇒ (ii): Each edge in G 2 is allowed, and thus Theorem 11(i) implies the equivalence. (ii) ⇒ (iii): Assume otherwise. By parity reasons, we have a set Y ⊆ V (G) with |Y | ≥ 2 such that f -odd(G − Y ) = f (Y ). So Y is an f -barrier, and is contained in a set S ∈ S with |S| ≥ 2 by Theorem 11, which contradicts (ii). (iii) ⇒ (i): Assume that G is not f -bicritical. Let g = f − χ{u,v} . 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