Let’s begin here with the 5-foot-9 elephant in the room. If you were to pick one player in the NHL who would most make the Islanders better outside of Connor McDavid, Auston Matthews and Leon Draisaitl, there would be at least a reasonable case for Johnny Gaudreau.

The Calgary winger is coming off a career-best season in which he scored 40 goals for the first time, breached 100 points for the first time — ranking second in the league with 115 — and was part of arguably the best line in hockey, alongside Elias Lindholm and Matthew Tkachuk.

Gaudreau — who grew up in Carney’s Point, just a few hours drive from UBS Arena — is hitting unrestricted free agency for the first time in his career, having reportedly turned down an offer of roughly eight years at over $10 million per year, informing the Flames that he would be leaving Calgary.

The Islanders are in desperate need of a winger who can maximize Mathew Barzal’s talent following a season in which their star center’s highlight reel consisted mostly of skating around the offensive zone, searching in vain for someone — anyone — to feed the puck to in a scoring position. Gaudreau would be that player.

The pieces do fit. But there is that pesky matter of the salary cap.

The Islanders currently have approximately $11.1 million in space, per CapFriendly. But that is before re-signing Noah Dobson, Alexander Romanov and Kieffer Bellows — all restricted free agents who the team qualified ahead of Monday’s deadline. In particular, Dobson, coming off a breakout season, and Romanov, who the team gave up the 13th pick in last week’s draft to acquire, will take a bite out of that number.

General manager Lou Lamoriello must also factor in an eventual contract extension for Barzal, whose $7 million cap hit will likely increase when his deal expires at the end of next season.

Gaudreau would surely command a maximum $11 million AAV on the open market, or close to it. So, to make a long story short, the Islanders would need to move some pieces around to make signing him possible.

For a player of Gaudreau’s caliber, though, you do what you need to do. The Flyers, who bought out Oskar Lindblom on Tuesday morning and are reportedly trying to offload James van Riemsdyk’s $7 million cap hit, seem to already be in the mode of clearing space to try and land the South Jersey native.

Anthony Beauvillier would be the most logical player for the Isles to try and move, but with a $4.15 million cap hit, that might not be enough on its own to clear the requisite space. In such a scenario, it could make sense to move Ross Johnston ($1.1 million).

If the Islanders cannot land free agency’s white whale — and it does seem an uphill battle — the ultimate need on the roster doesn’t change. Exactly where Lamoriello would look in this free-agent class is anyone’s guess, though he might not limit himself to free agency. J.T. Miller would be a fit, and returning to trade talks with Vancouver, which reportedly broke down on the draft floor in Montreal, might end up being the best option.

Otherwise, among available wingers, Toronto’s Ilya Mikheyev is coming off a 20-goal season and has a relationship with newly hired European scout Jim Paliafito, who scouted him extensively in Russia. Florida’s Mason Marchment, who Lamoriello signed as an undrafted free agent with the Maple Leafs in 2018, is coming off a breakout season in which he scored 47 points and has the sort of two-way game Lamoriello is known to like in his forwards.

There is nobody, though, who would impact the Islanders like Gaudreau.