I just want to ask can we increase backlinks to our site by putting 1x1 pixel image on our clients' sites or can anybody suggest a better solution?
<img src="http://www.mysite.com/" alt='http://www.mysite.com/' />
Google usually check for this kind of stuff and will get you penalized pretty fast. Place a normal link or a decent-sized image or don't place it at all.
The Year is 2013 and SEO isn't what it used to be
Since you have failed to mention the nature of your business and that of your clients', it's a guessing game on my part so it's hard to tell you for sure if any link would be valuable or not.
I'm assuming that you're a website designer and if that is the case then it's extremely unlikely that your client's site is also a website designing company. Which means that yes, you will gain Page Rank from general text or anchor links, but please note that Page Rank in this day and age has little to do with rankings.
Google and other search engines now work on very clever algorithms which know who you're linking to and who they are linking to. Google for instance prefers sites linking to their own interests as this is extremely natural. It is not natural for a site that is about cats to link to a site about buying and selling houses, for example, so with this said, hardly any juice is passed. It used to work but now it's all about the quality of links i.e. how relevant they are.
Links that are Best to Avoid
Footer links and aside links are the ones everyone should be trying to avoid unless they are completely 100% relevant to your site and theirs. These can be harmful because of Panda and Penguin; of course one or two may not hurt but is it worth the risk? Get links that help your direct traffic and your rankings.
Bad for You, Bad for your Client
As you're most likely aware, Google works on kind of an authority system and works out which sites are relevant to what the searcher is searching for. Basically, a good site about cats will have other websites linking to it from other related cat sites as well as sites linking to sites about cats... It does not help your client or yourself if your links are nothing to do with one another, in fact it can be bad. Of course one link may not hurt but generally it's water you shouldn't even bother getting into today.
However with this said, it can help your business and your client if you link to one another when you are in the same location as this can improve Local Rankings Via Citations or Links. So, for example, if their business is in Bournemouth, Dorset, UK and yours is as well, this may help your local rankings as you're vouching for one another which generally is a normal thing to do regardless of the topic.
Cloaking of Links
Google does not favour cloaking links, there are hundreds of ways how this can be done and again this is water you do not want to get into.
This includes some usual things such as:
AGAIN, DO NOT DO THIS.
THE BEST SEO... WELL IT'S NOT REALLY SEO
If you want customers, then rather than planting links on your customer's site you should be focusing on providing amazing customer service and delivering beyond their expectations. This is by far better than silly little links as you will gain revenues from them returning to you on a regular basis for more work as well as recommending your company as word of month which is golden and far superior to links.
Look after your customer and don't rely on them to pass the buck back. If you want one link from their site and they agree, make it legit and make it on one page. You could even have it on the about page but again... It doesn't really help your rankings at all unless their website is on topic with your own or they live in the same area.