On the existence problem of the total domination vertex critical graphs
Moo Young Sohn, Dongseok Kim, Young Soo Kwon, Jaeun Lee

TL;DR
This paper investigates the existence conditions of total domination vertex critical graphs based on the total domination number and maximum degree, providing a comprehensive classification and resolving several open cases.
Contribution
It establishes existence criteria for m-gamma_t-critical graphs based on parity and degree, and addresses previously unresolved cases for specific parameters.
Findings
No m-gamma_t-critical graphs for even m except m=4 when Delta is odd.
Existence of m-gamma_t-critical graphs depends on inequalities involving m and Delta.
Certain open cases remain unresolved, specifically when Delta equals 2*floor((m-1)/2) plus small odd integers.
Abstract
The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total domination number m and the maximum degree Delta : for even m except m=4, there is no m-gamma_t-critical graph regardless of the parity of Delta; for m=4 or odd m \ge 3 and for even Delta, an m-gamma_t-critical graph exists if and only if Delta \ge 2 \lfloor \frac{m-1}{2}\rfloor; for m=4 or odd m \ge 3 and for odd Delta, if Delta \ge 2\lfloor \frac{m-1}{2}\rfloor +7, then m-gamma_t-critical graphs exist, if Delta < 2\lfloor \frac{m-1}{2}\rfloor, then m-gamma_t-critical graphs do not exist. The only remaining open cases are Delta = 2\lfloor \frac{m-1}{2}\rfloor +k, k=1, 3, 5. Second, we study these remaining open cases when m=4 or odd m \ge 9. As the…
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Graph theory and applications
