How to check if a file name contains directory info?

I would like to check if a file name has any directory information within it, preferably without using a system dependent hack like index($file_name,'/')!=-1. I know of the File::Spec module, but can't seem to find a way to use that module to make this determination. Perhaps another module would work better. Here are some examples of file names:

# This does not have directory info, so the test should fail (i.e., return false)
my $file_name1='abc.txt';

# These do have directory info, so the test should succeed (i.e., return true)
my $file_name2='dir/abc.txt';
my $file_name3='/home/me/abc.txt';
my $file_name4='~me/abc.txt';

splitdir will return a list. Evaluated in scalar context, it returns the number of items in the list. If there is more than 1 item in the list, then you know there is a directory separator.

use warnings;
use strict;
use File::Spec qw();

while (<DATA>) {
    chomp;
    if (File::Spec->splitdir($_) > 1) {
        print 'PASS';
    }
    else {
        print 'FAIL';
    }
    print ": $_\n";
}

__DATA__
abc.txt
dir/abc.txt
/home/me/abc.txt
~me/abc.txt

Prints:

FAIL: abc.txt
PASS: dir/abc.txt
PASS: /home/me/abc.txt
PASS: ~me/abc.txt