Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Game of Arrows on 3-Legged Spider Graphs   The Game of Cycles is a combinatorial game introduced by Francis Su in 2020
in which players take turns marking arrows on the edges of a simple plane
graph. Su and his collaborators found winning strategies for game boards
including cycle graphs with chords and graphs with certain types of symmetry.
Their paper mentions the possible use of the Sprague-Grundy Theorem to analyze
the Game of Cycles, but cautions that "this is not a hopeful approach because
the size of the game tree generally grows exponentially with the number of
edges of the board." In this paper, we employ the Sprague-Grundy Theorem to
show that player two has a winning strategy for the Game of Cycles on any
3-legged spider graph with legs of odd length. Our proof leans heavily on a
notion of state isomorphism that allows us to decompose a game state into
states of smaller pieces of a graph, leading to nim-sum calculations with
Grundy values. Because the cycle cell victory condition is extraneous for tree
graphs (including spiders), we drop it from the rules and call the result the
Game of Arrows.

25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics