I would convert your segment coordinates to points, this way points that are not unique would show that the segment is not an end point.
x = [1,2,2,2,1;2,3,2,1,1];
y = [1,1,1,2,2;1,1,2,2,1];
plot(x,y,'-x')
xlim([0,3])
ylim([0,3])
axis equal
%converts to points with x,y coordinates
points = [x(1,:),x(2,:);y(1,:),y(2,:)];
%initialize test array
repeat_arr = zeros(size(points,2),size(points,2));
for i_points = 1:size(points,2)
%find if a point is repeated excluding itself
repeat_arr(i_points,[1:i_points-1,i_points+1:end]) = all(points(:,i_points)==points(:,[1:i_points-1,i_points+1:end]),1) ;
end
% now we know for each points if it is repeated
temp = any(repeat_arr,1);
%Reverse the segment to point process so we can mask any segment that does not have a repeated point
edge_mask = all([temp(1:end/2); temp(end/2+1:end)]);
%Create X Y array of segment using out mask
x_edges = x(:,edge_mask);
y_edges = y(:,edge_mask);
plot(x_edges,y_edges,'-x')
xlim([0,3])
ylim([0,3])
axis equal
