Characterizing factor critical graphs and an algorithm

In this paper, we show a necessary and sufficient condition which characterizes all factor critical graphs. Using this necessary and sufficient condition, we develop a linear time algorithm to determine whether a graph is factor critical if one of its maximum matchings is given. 1 Terminology and introduction All graphs considered in this paper are undirected, finite and simple. In general, we follow the terminology of [1]. Let G be a connected graph and w be a vertex not in V (G). Then H = G + w denotes the graph with vertex set V (H) = V (G)∪{w} and edge set E(H) = E(G)∪ {vw | v ∈ V (G)}. A graph is said to be factor critical if G−u has a perfect matching for any vertex u in G. A graph G is said to be bicritical if G − {u, v} has a perfect matching for any two vertices u and v in G. A matching M is called near perfect matching in G if all vertices but one are incident with edges in M . The vertex u not incident with any edge in M is said to be M -unsaturated. The concept of factor critical graphs is also generalized to n-critical graphs. Let G be a connected graph with ν vertices and n be an integer such that 0 ≤ n ≤ ν − 2 and n ≡ ν (mod 2). Then G is said to be n-critical if, for any subset S ⊆ V (G) with |S| = n, G− S has a perfect matching. There has been some research on this topic (see [4–9]). In [12], Yu gives a Tutte style necessary and sufficient condition for n-critical graphs. However, it does not help to design an efficient algorithm to determine n-critical graphs. In [8], Lou and Zhong give a necessary and sufficient condition which characterizes all bicritical graphs and develop an algorithm to determine whether a graph is bicritical using the condition. In this paper, we give a necessary and sufficient condition to characterize all factor critical graphs. Using our necessary and sufficient condition, we can design a linear time algorithm to determine all factor critical graphs. 52 DINGJUN LOU AND DONGNING RAO 2 A necessary and sufficient condition In this section, we give a necessary and sufficient condition for the factor critical graphs. It serves as a basis for the algorithm in Section 3. First, we give a lemma. Lemma 1: (Lou and Zhong [8]) Let G be a graph with a perfect matching M0. Then the following propositions are equivalent: 1. G is bicritical; 2. For any perfect matching M and any two different vertices x and y in G, there is an M -alternating path between x and y, which starts and ends with edges in E(G) \M ; 3. For every pair of vertices x and y in G, there is an M0-alternating path between x and y, which starts and ends with edges in E(G) \M0. Now we give a theorem which shows the relation between n-critical graphs and (n+ 1)-critical graphs. Theorem 2: Let G be a connected graph and w be a vertex not in V (G). Then G is n-critical if and only if G+ w is (n+ 1)-critical. Proof. We prove necessity first. Suppose G is n-critical. Then, for any subset S ⊆ V (G) such that |S| = n, G − S has a perfect matching. Let G′ = G + w. Let S′ ⊆ V (G′) such that |S′| = n+ 1. If w ∈ S′, then S1 = S′ \ {w} ⊆ V (G) such that |S1| = n, so G′ − S′ = G− S1 has a perfect matching. If w / ∈ S′ , let x ∈ S′ and let S2 = S ′ \ {x}. Then S2 ⊆ V (G) and |S2| = n, and G − S2 has a perfect matching M1. Assume xy ∈ M1. Then G− S′ = (G− S2)− {x} has a near perfect matching M2 = M1 \ {xy}. Since w is adjacent to every vertex of G, wy ∈ E(G′). So G′ − S′ has a perfect matching M2 ∪ {yw}. Hence G′ is (n+ 1)-critical. Now we prove sufficiency. Suppose G′ = G+ w is (n+ 1)-critical. Then, for any S′ ⊆ V (G′) with |S′| = n + 1, G′ − S′ has a perfect matching. Let S be any subset of V (G) such that |S| = n and let S′ = S ∪ {w}. Then S′ ⊆ V (G′) and |S′| = n+1. Since G′ is (n + 1)-critical, G − S = G′ − S′ has a perfect matching. Hence G is n-critical. Theorem 3: Let G be a graph with odd order, M be a near perfect matching in G, and u be the M -unsaturated vertex of G. Then G is factor critical if and only if, for any vertex v = u in G, there is a (u, v) M -alternating path Q such that Q starts and ends with edges in E(G) \M . Proof. First, we prove sufficiency. Suppose that, for any vertex v = u in G, there is a (u, v) M -alternating path Q such that Q starts and ends with edges in E(G) \M . We are going to prove that G is factor critical. Let w ∈ V (G) and G′ = G − w. If w = u then the original matching M is a perfect matching of G′. Otherwise, if w = u, since G has the near perfect matching CHARACTERIZING FACTOR CRITICAL GRAPHS 53 M and u is the only M -unsaturated vertex, then there must exists a vertex v such that wv ∈ M . By the hypothesis of this theorem, there is a (u, v) M -alternating path Q such that Q starts and ends with edges in E(G)\M . Notice that w does not belong to Q. Then M ′ = M E(Q) is a perfect matching of G′, where M E(Q) denotes the symmetric difference of M and E(Q). Then we prove necessity. Let G′ = G + w such that w / ∈ V (G). Obviously, G′ has a perfect matching M1 = M ∪ {wu}. Since G is factor critical, by Theorem 2, G′ is bicritical. By Lemma 1, we know that, in particular, there is a (u, v) M1-alternating path Q for each v = u ∈ V (G) such that Q starts and ends with edges in E(G′) \M1. Since uw ∈ M1 and Q is an M1-alternating path starting and ending with edges in E(G′) \M1, w / ∈ V (Q). Hence Q is a (u, v) M -alternating path in G such that Q starts and ends with edges in E(G) \M . The necessity is then proved. 3 Description of the algorithm In this section, we give a linear time algorithm to determine whether a connected graph G is factor-critical if one of its maximum matching is given. A near perfect matching M is an input to this algorithm. Hence the existence of M is tested prior to this algorithm. Moreover, M should not necessarily be a near perfect matching. The algorithm can accept a maximum matching as an input, and test if it is a near perfect matching. ALGORITHM: 1. Input a maximum matching M of G; // O(|E|) 2. IfM is not a near perfect matching, then RETURN(false); (G is not factor-critical) // O(|V |) 3. Else use Procedure 1 to construct an M -alternating tree from the M -unsaturated vertex u to find an M -alternating path P from u to v such that P starts and ends with edges in E(G) \M for every v = u in V (G); // O(|E|) 4. If for some v = u in V (G), there is not such a path P (Procedure 1 returns false), then RETURN(false); (G is not factor-critical) // O(1) 5. Otherwise, RETURN(true); (G is factor-critical). // O(1) Procedure 1 uses the idea of [11] for finding an M -augmenting path to build an M -alternating tree starting from u. But it finds M -alternating paths from u to every vertex v = u such that the paths start and end with edges in E(G) \ M . We give the algorithm of Procedure 1 in the following. 54 DINGJUN LOU AND DONGNING RAO Procedure 1: 1. Use BFS strategy to build an M -alternating tree T rooted at u. First, T := ∅; 2. Put u into the even vertex queue Q; Mark u even; 3. Repeatedly take the first vertex x from Q, do the following Steps 4−7 until Q is empty; 4. For each edge xy incident with x do the following Steps 5−7; 5. Case 1: y is not visited. Let yy′ ∈ M ; T := T ∪ {xy, yy′}; Mark y odd and y′ even; Put y′ into the queue Q; Set Pre(y) := x; Pre(y′) := y; End of Case 1; 6. Case 2: y is marked odd. We do nothing in this case; 7. Case 3: y is marked even. Track along the Pre chains from x and from y respectively until we find the first common ancestor t of x and y. That is, we find vertex sequences P = (x =) a1, a2, . . . , am(= t) and R = (y =) b1, b2, . . . , bn(= t) such that Pre(ai) = ai+1, i = 1, 2, . . . ,m − 1, Pre(bj) = bj+1, j = 1, 2, . . . , n − 1, and ai = bj unless i = m and j = n; For each ai (i = 1, 2, . . . ,m− 1), set Pre(ai) := t; if ai is marked odd, then mark ai even and put ai into the queue Q; For each bj (j = 1, 2, . . . , n− 1), set Pre(bj) := t; if bj is marked odd, then mark bj even and put bj into the queue Q; End of Case 3; 8. If all vertices of G are marked even, then RETURN(true); otherwise RETURN(false); Notice that, for any even vertex x in T , if xy ∈ M , then there is an M -alternating path from u to y in T such that P starts and ends with edges in E(G) \ M . So it suffices to check that every vertex (except u) is marked even after the execution of Steps 3−7 to determine that G is factor critical. It is equivalent that every matching edge xy in M lies in a blossom of T . In Procedure 1, Pre is an array. For each vertex v in G, Pre has an element Pre(v). If v is not in any blossom of T , then Pre(v) is the father of v in T . If v lies in a blossom of T , then Pre(v) is the root of a blossom containing v which has been processed. Here the terminology blossom and root of blossom comes from the classical paper of Edmonds [2]. CHARACTERIZING FACTOR CRITICAL GRAPHS 55 Procedure 1 only takes O(|E|) time since it uses BFS strategy to build the M alternating tree T . Notice that, in Step 7, if the algorithm tracked the vertex sequences P = (x =) a1, a2, . . . , am(= t) and R = (y =) b1, b2, . . . , bn(= t) once, then the algorithm will not track any subsequence of P and R of length at least 3 one more time because the algorithm has set Pre(ai) := t and Pre(bj) := t, i = 1, 2, . . . ,m−1, j = 1, 2, . . . , n − 1. So it takes at most O(|E| + |V |) time to process all blossoms. But we have assumed that G is a connected graph. So O(|E|) +O(|V |) = O(|E|) It is easy to see that the whole algorithm only takes O(|E|) time. It is a linear time algorithm and hence is optimal. However, by [11], it takes O(|V |1/2|E|) time to find the near perfect matching M in G. To determine whether a graph G is factor critical spends the main time in finding a near perfect matching in G. If we use the definition of factor critical graphs to design an algorithm, then we delete every vertex v and try to find a perfect matching in G − v. In this case, the algorithm needs O(|V |3/2|E|) time. So our algorithm has higher efficiency.