From endpoint order to deletion certificates
This demonstration follows Pedro M. M. de Castro, Minimum blockers and extremal graphs for nonnested matchings, arXiv:2609.14484v2 [math.CO], 8 October 2026. Its author is Pedro M. M. de Castro alone. Section and theorem references below refer to v2.
For edges {a, b} and {c, d} with a < c and distinct endpoints, nesting means a < c < d < b. Crossing means a < c < b < d, and is allowed. Unselected edges remain part of the graph. The drawings are schematic: endpoint order, not the particular shape of a curve, determines compatibility.
The classification on 2k vertices
The catalogue implements the three disjoint families of Theorem 2.1. In a left blocker L(m, a), the fan consists of {1, 2} through {1, m+1}. For each length j from m+1 through k, delete {a_j,a_j+j}, where the positions increase weakly from 2 through m+1. Reflect the order to obtain the right family. The mixed family is C_m in Equation (2.2), for 1 ≤ m ≤ k−2. The case k=1 is handled separately.
There are 2^(k−1) + 2^(k−1) + (k−2) = 2^k + k − 2 minimum blockers for k≥2. “Minimum” refers to cardinality, not merely the inability to remove another deletion.
What the certificate establishes
The implementation of Algorithm 1 uses its event sweep and its internal-split and end-split witness constructions. The input is a set D with |D|≤k on exactly 2k vertices. Every returned matching is checked for distinct endpoints, increasing right endpoints and disjointness from D before it is displayed. A blocker answer checks the parameters against the classified formulas.
The auxiliary blocks are B(i,j) = {{a,a+j−i}: 2i+1 ≤ a ≤ i+j}. Failure of a path from 0 to k only blocks this interval-block subfamily. It does not by itself establish that all nonnested matchings are blocked. The six-vertex example D={{1,2},{1,4},{5,6}} illustrates the difference: {{1,3},{2,5},{4,6}} survives. When more than k edges are deleted, the demo explicitly leaves the domain of Algorithm 1 and offers a separate finite enumeration.
Each answer also names the object that produced it: the surviving auxiliary path and its blocks, the matching P_p or ρ(P_p) of Equation (2.5) for an internal split, or the first length at which the monotone-extension condition of Lemma 2.5 fails for an end split. A blocker answer reports its reachable indices and split t from Lemma 2.3. The right family is written ρ(L(m,a)), with its positions read in the reflected order.
The computational certificate routine follows the quadratic-time construction in Theorem 3.2. Rendering a table of all interval blocks is additional display work and is not part of the theorem's auxiliary-storage claim. The demo does not claim a measured speedup or an optimal general matching algorithm.
The four deletion regions in v2
Theorem 4.5 permits integers a,b≥1, 0≤c≤min(a,b), w=a+b−c, k≥3w−1 and n≥N=2k+a+b−1. The demo validates these hypotheses, including the boundary where J and B meet at one vertex. The edge matrix shows pairs of graph vertices; it is not the 123-avoiding permutation-matrix embedding of Proposition 3.3.
Set p=k−b+c, q=k+a+w, h=a+w and f=a+2w−1. The left and right fans are Equation (4.5). The rectangle joins every vertex h…f to every vertex q…q+a+b−1. Triangle row i deletes {a−c+i,q+j} for 0≤j<i and 1≤i≤c. These four regions are disjoint. After the core N vertices, each new last vertex is adjacent exactly to the original vertices 1…k−1; no edges are added between new vertices.
With r=k−1 and ξ = ab − c(c+1)/2, the construction has e(G)=rn+ξ edges. Short edges have length at most r. Their count is (3r²+r)/2+ξ; the remaining rn−(3r²+r)/2 edges are long. The running example k=8,a=b=2,c=1,n=19 deletes 11+11+12+1=35 edges and retains 136, split as 80 short and 56 long.
The triangle example restores {2,13} and displays a valid target matching whose unused vertices are exactly 5,6,7. Once restored, that graph no longer avoids the target. Its higher edge count is consequently not presented as an extremal lower bound.
Construction quality versus the unknown optimum
For k≥5 and n≥2k+1, “Best parameters in this family” implements Corollary 4.8. It balances a,b and chooses the least triangular cost at the optimizing total. It does not find the global extremal graph. The plateau of equally good parameter totals can give multiple maximizers.
The upper bound (k−1)n+binom(k−1,2) is Proposition 4.9. Near n=2k+1 the demo also uses Proposition 4.4, including its averaged bound when k≥10. The coefficient 1/18 and the optimized excess binom(floor((k+1)/3),2) describe this construction; the exact extremal value and best leading coefficient remain open in general.
The guided investigation and collection
The guide connects a selected matching, a surviving counterexample, a blocker by isolation, a classified minimum blocker and the coordinated construction. Each step states what to do and what evidence completes it. Starting a guide from an existing exploration saves a separate copy; leaving restores the graph, target, deletions, selection, applied construction and collection view, unless you choose to keep the guide graph. Reset during the guide resets only its current step and is undoable.
The collection always uses the current vertex count, target size and deleted edges. A target has exactly k pairs; it is perfect only when n = 2k. The default target follows floor(n/2), and the target controls can hold a smaller size fixed. Every page reports its shown range and full count. Six cards per page is a display limit, not a limit of six or five matchings. Blocked patterns can be inspected separately; selecting a surviving card changes only the matching selection.
Finite verification and interaction
The optional exact check enumerates each 2k-vertex subset and each Catalan nonnested perfect-matching pattern on it. Thus its target count is binom(n,2k) × binom(2k,k)/(k+1). It reports the number of surviving target matchings, not a maximum matching size. Enumeration runs in a cancelable worker, is limited to 2,000,000 target patterns and k≤9, and is invalidated when the graph changes. No result from an interrupted check is presented as complete.
Source tests additionally use an independent recursive pairing oracle for small cases and an independent enumeration by complementary endpoint positions for the 19-vertex example. Tests exhaust every size-k deletion set through k=5, exercise the algorithm's different witness branches and reflections, verify construction boundaries and counts, and compare optimization with brute force over feasible parameters.
Tap two vertices, or use the labeled vertex menus and the explicit action button. The same actions work with a keyboard. Matrix inspection has equivalent endpoint menus and an enlarged view with 40-pixel cells, native scrolling and row and column numbers that stay in view. Rejection feedback is placed next to the editing controls. Ordinary touch scrolling is preserved; diagrams do not capture swipe gestures. Undo and Redo retain up to 80 applied changes, including parameters and selected/deleted edges. Reset is undoable and also resets the region filter and inspected edge. The shared edge editor is contained in the active tab panel. This local preview does not persist edits across a page reload.
Credits and component licenses
Demo by Pedro M. M. de Castro. Derived from the single-author article cited above. Tools for computational exploration and implementation: OpenAI Codex and Anthropic Claude.
© 2026 Pedro M. M. de Castro, original contributions. Code: Apache 2.0. Original text and figures: CC BY 4.0. NOTICE · License scope · Site scope note. Third-party materials retain their own terms.
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