Matchingsmarked

Matchingsmarked is a term used in graph theory and related fields to describe a representation of a matching in a graph where the matched edges are explicitly marked. In a graph G=(V,E), a subset M⊆E is a matching if no two edges in M share a common vertex. In matchingsmarked representations, each edge e∈E is assigned a marking indicator s(e)∈{marked,unmarked} (often implemented as a binary mark or color). Edges with mark corresponding to M are considered part of the matching; unmarked edges lie outside M. This marking is used primarily for visualization, algorithm design, and formal proofs, as it makes the structure of the matching immediately apparent.

Construction and variants of the marking scheme are flexible. Common approaches include coloring matched edges with a distinct color, labeling them with an M, or placing a tick mark. In directed or bipartite graphs, the concept extends by marking directed edges or bipartite pairings that belong to a matching. When multiple matchings are analyzed simultaneously, multiple marks or colors can denote different edge sets M1, M2, and so on, enabling comparative visualization.

Applications and use cases include algorithm design and education. Matchingsmarked helps illustrate augmenting paths and steps in algorithms such as greedy matching or the blossom algorithm, and is often used in software tools for visualizing matchings or in lecture notes to clarify proofs.

See also: Matching (graph theory), Maximum matching, Edmonds' blossom algorithm, Graph visualization.