What does it mean to deny the Holy Ghost? If this is the worst possible sin, the answer to this question is crucially important. The Savior explained that anything spoken against the Son of Man would be forgiven, but whosoever speaketh against the Holy Ghost, it shall not be forgiven him, neither in this world, neither in the world to come (Matt 12:32). However, it is impossible to deny the Holy Ghost without also denying the Christ. Therefore, in order to qualify as a son of Perdition, the individual must deny both the Holy Ghost and the Savior, Having denied the Holy spirit after having received it, and having denied the Only Begotten Son of the Father, having crucified him unto themselves and put him to an open shame (DC 76:35).
Furthermore, we are to understand that in order to deny the Holy Ghost, the individual must have first had a strong testimony by the power of the Holy Ghost, '[The sons of perdition are] all those who know my power, and have been made partakers thereof, and suffered themselves through the power of the devil to be overcome, and to deny the truth and defy my power (DC 76:31).' Another key element comes from Alma. He states that the rebellion is a willful rebellion, if ye deny the Holy Ghost when it once has had place in you, and ye know that ye deny it.
The last element is that the sons of perdition must "endure to the end." In other words, they continue in sin because it is impossible for them to repent (Heb 6:4-6). Then, at the last day, they are found filthy still, That which breaketh a law, and abideth not by law, but seeketh to become a law unto itself, and willeth to abide in sin, and altogether abideth in sin, cannot be sanctified by law, neither by mercy, justice, nor judgment. Therefore, they must remain filthy still (DC 88:35).
Joseph Smith
"All sins shall be forgiven, except the sin against the Holy Ghost; for Jesus will save all except the sons of perdition. What must a man do to commit the unpardonable sin? He must receive the Holy Ghost, have the heavens opened unto him, and know God, and then sin against Him. After a man has sinned against the Holy Ghost, there is no repentance for him. He has got to say that the sun does not shine while he sees it; he has got to deny Jesus Christ when the heavens have been opened unto him, and to deny the plan of salvation with his eyes open to the truth of it; and from that time he begins to be an enemy." (Teachings of the Prophet Joseph Smith, p. 358 as taken from Latter-day Commentary on the Book of Mormon compiled by K. Douglas Bassett, p. 333)
Joseph Fielding Smith
"All who partake of this, the greatest of sins, sell themselves as did Cain to Lucifer. They learn to hate the truth with an eternal hatred, and they learn to love wickedness. They reach a condition where they will not and cannot repent. The spirit of murder fills their hearts and they would, if they had the power, crucify our Lord again, which they virtually do by fighting his work and seeking to destroy it and his prophets." (Doctrines of Salvation, vol. 1, p. 49)