RickyDemer/ Sandbox

I consider endgame play in extremely small, fully-enclosed regions

i.e., each outside intersection that is adjacent to at least one of the region's intersections has an assumed-immortal stone

that do not trivially split into more than one region in the way I describe near the end of this lede.

Such regions are determined by the structure ( [ext] https://en.wikipedia.org/wiki/Graph_isomorphism) of the adjacency relation on

{ immortal black stone , immortal white stone } union its set of other intersections

which results from having there be no edge between the immortal stones.

For this article, I do allow isomorphisms to swap colors.

Trivial Splitting:

Edges between intersections that are both adjacent to both immortal stones are irrelevant because

Such an edge affects neither suicideness nor other captures, since reaching either of the edge's intersections through same-color stones means reaching an immortal same-color stone through same-color stones without needing the edge.

and

Such an edge does not affect territory, since reaching either of the edge's intersections through empty intersections means reaching both immortal stones through empty intersections without needing the edge.

.

Thus, if the intersections without immortal stones are not all connected via edges who's intersections aren't both adjacent to both immortal stones , then the region splits into the connected components of those intersections with those edges .

(0) Accordingly,

(a) for all stones on the boundaries of this page's diagrams, if the stone is not shown as on a board-edge then the stone is assumed to be immortal

and

(b) cross s represent immortal stones whose color does not matter

.

By "colored region" , I mean region together with a legal coloring of its inside , in the sense that does include colorings which create one or more new immortal stones in a way that makes the new region trivially split as described above.

For example, for this region,

this is one of the corresponding colored regions, even though it has a new immortal stone and trivially splits.

(1) I use territory scoring with 1-stone suicide prohibited.

(2) I assume that if multi-stone suicide is allowed, then with the possible exception of the stones placed as moves that are suicide, the contribution of captures to the score depends only on what stones were captured, rather than instead-or-also on who did the capturing.

(i.e., that even if white+circle is allowed, playing it cannot help White.)

(3) I do not assume that the players are treated the same regarding suicide. (For example, my results apply even to hypothetical rulesets that allow multi-stone suicide by White and prohibit suicide by Black.)

(4) Except as specified in (5), I assume that either

(a) there is no encore and if play ends with a ko mouth such that take-followed-by-fill would make the 2 new stones immortal, then that mouth's stone is scored as dead and that mouth's empty intersection is not scored as territory for that stone's player

(b) the effect of pass on ko bans and the external situation (ko threats and possibly dame) are such that (a) might-as-well apply .

( i.e., that except as described in (5), Black getting both black+circle s for free cannot h??? Black. )

(5) For ties that remain after considering ko threats, I use removing (4)'s assumption as the next tiebreaker.

(Even if one ignores territory-scoring rules for which 4(a) does not necessarily apply, this gives a nod to area-scoring.)

Due to 6(a), 4(b) is weaker than it might seem.

(6) The regions I consider are all connected and small-enough that, for all corresponding colored regions,

(a) There are at most finitely-many ko threats, even in the sense in which sending 2 receiving 1 qualifies as infinitely many.

and

(b) All ko threats can trivially be removed in a way which would not cost any points if done in an encore. (In particular, they have no non-removable ko threats.)

and

(c) For all at-all-loss-making ko threats, the threatener has at least one alternative ko threat which is strictly better and not-at-all loss-making.

and

(d) There are no board plays inside which can be at-all relevant to a loop that includes at least one board play inside .

.

(short summary of #6: finitely-many removable threats which are lossless and not local)

(7) Except as specified in (1) and (2) and (4), I make no assumptions on what sort of territory scoring is used.

(8) I use ko threats as tiebreaker for sequences of 1 or more board plays that are both canonical and otherwise-equivalent , and mention them or the lack thereof whenever the colored region is settled.

[edit]