Col and its myopic cousin
Blue (Left) and Red (Right) take turns colouring a vertex. You may not colour a vertex next to one of your own colour. Whoever cannot move loses. In Col a vertex looks at all its neighbours; in Myopic Col the board is a linked list or a binary tree and a node looks only along its arrows, at its children. Every position has an exact value, always a number or a number plus * (cold). A node nobody looks at, whose children are uncoloured, is ringed: it splits off as a lone *, which is the 2016 conjecture on binary trees and, as measured here, much more. With hints on, every legal move is marked ✓ (keeps the win) or ✗.
value of the position
components
How the values are computed
Both games are one structure: each vertex lists the vertices it looks at and may not take a colour one of them has. A coloured vertex leaves the game and forbids its colour on everyone looking at it; a vertex with no colour left is dead, and an arc whose ends no longer share an allowed colour is dropped. What is left falls apart into components, the game is their sum, and each component is valued once from its options and reduced to canonical form (trees are keyed up to isomorphism).
Checked against the literature
M. Bovee, K. Burke and C. Tennenhouse, Games from Basic Data Structures (arXiv 1605.06327) introduce Myopic Col. Their lemma on lists (n·*, (n−1)·* ∓ 1) holds for n ≤ 400 and their a·* + b − c formula on every colouring of lists up to 13 nodes and of list forests up to 9. Their conjecture, G(T) = * + G(T₁) + G(T₂) when the root and its two children are uncoloured, holds on all 1,883,902 cases among the 31,469,089 positions of binary trees with up to 11 nodes; the same split holds for roots with one or many children and for any vertex of any digraph that nobody looks at. Conway's theorem that Col is cold is checked on every colouring of every connected graph up to 8 vertices, and Myopic Col turns out cold everywhere it was measured too.