The　3-Decomposition　Conjecture　states　that　every　connected　cubic　graph　can　be　decomposed　into　a　spanning　tree,　a　set　of　cycles,　and　a　matching.　It　has　been　proved　independently　by　different　groups　of　people　that　every　connected　cubic　graph　can　be　decomposed　into　a　spanning　tree,　a　set　of　cycles,　and　a　set　of　vertex-disjoint　paths　of　at　most　two　edges.　In　this　paper,　we　establish　a　bound　on　the　number　of　paths　of　two　edges,　proving　that　every　connected　cubic　graph　on　n　vertices　can　be　decomposed　into　a　spanning　tree,　a　set　of　cycles,　and　a　set　of　vertex-disjoint　paths　of　at　most　two　edges　such　that　the　number　of　paths　of　two　edges　is　at　most　n-4　6　.　Our　proof　is　based　on　a　structural　analysis,　which　might　provide　a　new　approach　to　attack　the　3-Decomposition　Conjecture.　(c)　2025　Elsevier　B.V.　All　rights　are　reserved,　including　those　for　text　and　data　mining,　AI　training,　and　similar　technologies.