An　even　cycle　C　of　a　graph　G　is　nice　if　the　graph　G　-　V(C)　has　a　perfect　matching.　An　orientation　of　G　is　a　Pfaffian　orientation　if　every　nice　cycle　of　G　has　an　odd　number　of　edges　directed　in　either　direction　of　the　cycle.　A　graph　is　Pfaffian　if　it　has　a　Pfaffian　orientation.　If　a　graph　is　Pfaffian,　then　the　number　of　perfect　matchings　of　it　can　be　computed　in　polynomial　time.　In　this　paper,　we　focus　on　a　special　type　of　1-extendable　bipartite　graph　with　maximum　degree　Delta(G)　=　vertical　bar　V(G)vertical　bar/2.　We　characterize　some　properties　of　Pfaffian　graphs　in　this　type.　According　to　the　properties,　we　find　an　algorithm　in　time　O(vertical　bar　E(G)vertical　bar(2))　to　determine　whether　a　graph　G　in　this　type　is　Pfaffian　or　not.　Furthermore,　if　G　is　Pfaffian,　this　algorithm　also　constructs　a　Pfaffian　orientation　of　it.　(C)　2014　Elsevier　B.V.　All　rights　reserved.