The Bottleneck Lemma states that if a bipartite graph has a bottleneck, then no perfect matching exists. This is considered obvious from the counting argument.
Conditions: There is a subset S of the left-side vertices.; |S| > |E(S)|.
The Bottleneck Lemma states that if a bipartite graph has a bottleneck, then no perfect matching exists. This is considered obvious from the counting argument.
Conditions: There is a subset S of the left-side vertices.; |S| > |E(S)|.
A bottleneck is defined as a subset S of the left-side vertices whose neighborhood on the right side is strictly smaller than S itself. Mathematically, S is a bottleneck if and only if |S| > |E(S)|.
Conditions: S is a subset of the left-side vertices.; E(S) is the neighborhood of S in the right-side set.
A bottleneck is defined as a subset S of the left-side vertices whose neighborhood on the right side is strictly smaller than S itself. Mathematically, S is a bottleneck if and only if |S| > |E(S)|.
Conditions: S is a subset of the left-side vertices.; E(S) is the neighborhood of S in the right-side set.
The example proves no matching is possible because the highlighted subset of three girls has a neighborhood of only two boys. Since |S| = 3>2 = |E(S)|, this subset is a bottleneck, and by the Bottleneck Lemma, no matching can cover all girls.
Conditions: A bipartite graph with left set G and right set B.; A highlighted subset S of three girls.; Their collective neighborhood E(S) contains only two boys.
The example proves no matching is possible because the highlighted subset of three girls has a neighborhood of only two boys. Since |S| = 3>2 = |E(S)|, this subset is a bottleneck, and by the Bottleneck Lemma, no matching can cover all girls.
Conditions: A bipartite graph with left set G and right set B.; A highlighted subset S of three girls.; Their collective neighborhood E(S) contains only two boys.