#!/usr/bin/perl

# https://perlmonks.org/?node_id=1217048

use strict;
use warnings;

local $_ = do { local $/; <DATA> };

/^(.*)\\.*\\.*\n(?:\1\\.*\\.*\n)*$/ and s/\Q$1\E//g;

print;

__DATA__
C:\somedirectory\anotherDirectory\WorkingDirectory\myFile.txt
C:\somedirectory\anotherDirectory\WorkingDirectory\Logs\myLogs.txt
C:\somedirectory\anotherDirectory\WorkingDirectory\Stuff\myStuff.txt