Matching Extendabilities of G = Cm ∨ Pn

A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G = C m ∨ P n ( m ≥ 3 and n ≥ 1 ) be the graph join of C m (the cycle with m vertices) and P n (the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable.

Metric Dimension of Graph Join P2 and Pt

The following metric dimension of join two paths $P_2 + P_t$ is determined as follows. For every $k = 1, 2, 3, ...$ and $t = 2 + 5k$ or $t = 3 + 5k$, the dimension of $P_2 + P_t$ is $2 + 2k$ whereas for $t = 4 + 5k, t = 5(k+1)$ or $t = 1 + 5(k+1)$, the dimension is $3 + 2k$. In case $t \geq 7$, the dimension is determined by a chosen (maximal) ordered basis for $P_2 + P_t$ in which the integers 1, 2 are the two consecutive vertices of $P_2$ and the next integers $3, 4, ..., t + 2$ are the $t$ consecutive vertices of $P_t$. If $t \geq 10$, the ordered binary string contains repeated substrings of length 5. For $t 7$, the dimension is easily found using a computer search, or even just using hand computations.