A theta graph is denoted by \(\theta(a,b,c)\), where \(a \leq b \leq c\). It is obtained by subdividing the edges of the multigraph consisting of \(3\) parallel edges \(a\) times, \(b\) times, and \(c\) times each. In this paper, we show that the theta graph is matching unique when \(a \geq 2\) or \(a = 0\), and all theta graphs are matching equivalent when only one of the edges is subdivided one time. We also completely characterize the relation between the largest matching root \(\alpha\) and the length of path \(a, b, c\) of a theta graph, and determine the extremal theta graphs.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.