  This paper reports on the details of the International Competition on Graph
Counting Algorithms (ICGCA) held in 2023. The graph counting problem is to
count the subgraphs satisfying specified constraints on a given graph. The
problem belongs to #P-complete, a computationally tough class. Since many
essential systems in modern society, e.g., infrastructure networks, are often
represented as graphs, graph counting algorithms are a key technology to
efficiently scan all the subgraphs representing the feasible states of the
system. In the ICGCA, contestants were asked to count the paths on a graph
under a length constraint. The benchmark set included 150 challenging
instances, emphasizing graphs resembling infrastructure networks. Eleven
solvers were submitted and ranked by the number of benchmarks correctly solved
within a time limit. The winning solver, TLDC, was designed based on three
fundamental approaches: backtracking search, dynamic programming, and model
counting or #SAT (a counting version of Boolean satisfiability). Detailed
analyses show that each approach has its own strengths, and one approach is
unlikely to dominate the others. The codes and papers of the participating
solvers are available: https://afsa.jp/icgca/.
