Let　G　be　a　connected　simple　graph　on　n　vertices.　Gallai＇s　conjecture　asserts　that　the　edges　of　G　can　be　decomposed　into　⌈n/2⌉　paths.　Let　H　be　the　subgraph　induced　by　the　vertices　of　even　degree　in　G.　Lovász　showed　that　the　conjecture　is　true　if　H　contains　at　most　one　vertex.　Extending　Lovász＇s　result,　Pyber　proved　that　the　conjecture　is　true　if　H　is　a　forest.　A　forest　can　be　regarded　as　a　graph　in　which　each　block　is　an　isolated　vertex　or　a　single　edge　(and　so　each　block　has　maximum　degree　at　most　1).　In　this　paper,　we　show　that　the　conjecture　is　true　if　H　can　be　obtained　from　the　emptyset　by　a　series　of　so-defined　α-operations.　As　a　corollary,　the　conjecture　is　true　if　each　block　of　H　is　a　triangle-free　graph　of　maximum　degree　at　most　3.　©　2004　Elsevier　Inc.　All　rights　reserved.