Network　Function　Virtualization　(NFV),　which　decouples　network　functions　from　hardware　and　transforms　them　into　hardware-independent　Virtual　Network　Functions　(VNF),　is　a　crucial　technology　for　many　emerging　networking　domains,　such　as　5G,　edge　computing　and　data-center　network.　Service　Function　Chaining　(SFC)　is　the　ordered　set　of　VNFs.　The　VNF　deployment　problem　is　to　find　the　optimal　deployment　strategy　of　VNFs　in　SFC　while　guaranteeing　the　Service-Level　Agreements　(SLAB).　Existing　VNF　deployment　researches　mainly　focus　on　sequences　of　VNFs　without　energy　consideration.　However,　with　the　rapid　development　of　application　requirement,　the　SFCs　evolve　from　sequence　to　dynamic　graph　and　the　service　providers　become　more　and　more　sensitive　to　the　energy　consumption　in　NFV.　Therefore,　in　this　paper,　we　identify　the　Energy-efficient　Graph-structured　SFC　problem　(EG-SFC)　and　formulate　it　as　a　Combinatorial　Optimization　Problem　(COP).　Benefiting　from　the　recent　advances　in　machine　learning　for　COP,　we　propose　an　end-toend　Graph　Neural　Network　(GNN)　based　on　constrained　Deep　Reinforcement　Learning　(DRL)　method　to　solve　EG-SFC.　Our　method　leverages　the　Graph　Convolutional　Network　(GCN)　to　represent　the　Q-network　of　Double　Deep　Q-Network　(DDQN)　in　DRL.　The　mask　mechanism　is　proposed　to　deal　with　the　resources　constraints　in　COP.　The　experimental　results　show　that　the　proposed　method　can　deal　with　unseen　SFC　graphs　and　achieve　better　performances　than　greedy　algorithm　and　traditional　DDQN.