Some Visualforce and Apex experimentation with rerendering

RerenderNoAction.cls

public with sharing class RerenderNoAction {
    public RerenderNoAction() {
        viewMeCnt = 0;
    }
    
    public Integer viewMeCnt { get; set; }
    
    public String parentOpt { get; set; }
    public List<SelectOption> getParentOptions() {
        List<SelectOption> opts = new List<SelectOption>();
        opts.add(new SelectOption('',''));
        opts.add(new SelectOption('1', '2'));
        opts.add(new SelectOption('3', '4'));
        return opts;
    }
    
    public String opt { get; set; }
    public List<SelectOption> getOptions() {
        viewMeCnt = viewMeCnt + 1;
        List<SelectOption> opts = new List<SelectOption>();
       
        if (parentOpt == '1') {
            opts.add(new SelectOption('1111', '2222'));
            opts.add(new SelectOption('3333', '4444'));
        } else {
            opts.add(new SelectOption('7777', '8888'));
            opts.add(new SelectOption('9999', '0000'));
        }
        return opts;
    }
}

rerenderNoAction.page

<apex:page controller="RerenderNoAction" showheader="false" sidebar="false" standardStylesheets="false">
    <apex:form >
  
        <apex:actionRegion >
            <apex:selectList value="{!parentOpt}" multiselect="false" size="1">
                <apex:selectOptions value="{!parentOptions}"/>
                <apex:actionsupport event="onchange" rerender="panel" />
            </apex:selectList>
        </apex:actionRegion>  
        <br />
        <apex:outputPanel id="panel">
            Here we are!<br/>
            View Count: {!viewMeCnt}<br />
              
            <apex:selectList value="{!opt}" multiselect="false" size="1" required="true" id="opt">
                <apex:selectOptions value="{!options}" />
            </apex:selectList>
        </apex:outputPanel>
        
    </apex:form>
</apex:page>